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Video tags
Subtitles
Russian
00:00:04
you give up. Good, normal, great00:00:06
I hope that everything will be fine. Everything will be fine00:00:08
without interruptions, as if everything should be00:00:10
Okay, because the Internet is good and has everything00:00:12
be well today tell me How00:00:15
your mood How you feel about classes00:00:16
if they ask if the topic is difficult, see immediately00:00:19
I say that the topic is not difficult for me very often00:00:22
When I ask you followers there in00:00:25
Telegram or Instagram, what are the topics there00:00:27
for you are the most difficult of which you00:00:29
I would like some webinars there00:00:30
free something often write a topic00:00:33
I'm a derivative of it. To be honest, I don't like it00:00:35
about you free webinar ago00:00:37
that the topic is easy, in fact there is nothing to talk about00:00:40
we just have to tell the truth00:00:42
the plate you use and00:00:44
find all the derivatives there really like00:00:46
to explain something to the teacher Well, there is nothing there00:00:50
there is a sign you must be able to use it00:00:53
use the fact that you have this one00:00:54
the sign will be on the supplementary examination00:00:56
materials, i.e. you don't even need it there00:00:58
cramming to understand somehow She is deep there00:01:00
it will be before your eyes, you just need to00:01:02
to learn00:01:04
take into account how to use it and that's it00:01:07
i.e., there is nothing so extravagant here00:01:09
Today I will tell you what it is00:01:12
why is it so derivative?00:01:13
is used, let's look at the plate00:01:16
there are some nuances, I will also tell you about them00:01:19
I'll tell you, let's practice and how it will be00:01:21
dismiss the class that will be today00:01:22
necessarily only at the end of the lesson in00:01:25
principles Well, as always, but on nmte so00:01:28
see Last year if this one happens00:01:31
the situation that there will be an MRI is in Last year00:01:32
she will not give additional ones either00:01:34
materials and they were absolutely00:01:35
identical to those that were given at the final exam in00:01:38
principles without difference all the same to such00:01:40
materials are given and they are basically nothing00:01:42
not different Well, that's the story00:01:44
Okay, let's be together a little bit00:01:47
you to start and move towards ours00:01:49
topic, we talk about the derivative and about it00:01:52
geometric and mechanical content first00:01:55
the first thing I want to tell you is the First00:01:58
what I want to talk about Today is what00:02:00
usually start this topic in general00:02:03
on the course, this is the story. That's it00:02:05
look at what 100% of them told you at school00:02:09
such a derivative Yes and 100% you sooner than00:02:11
everyone said today at half past one00:02:14
slide hours today is 34 hours00:02:17
one and a half There plus or minus everything is always in00:02:19
In principle, that is, we are on schedule with you00:02:22
Usually tell this story to you00:02:24
they told exactly what it was at school00:02:26
derivative and one hundred percent said the definition00:02:28
of such a plan that the derivative is a relation00:02:30
of the increment of the function to the increment of the argument00:02:32
well, I think that I have heard exactly that, too00:02:36
I will not ask you because I00:02:37
I understand that you are fine if fine00:02:40
I understand that you hardly understood it00:02:42
definition well increase And what is an increase00:02:45
And what is a function, what is an argument, that is00:02:46
many questions arise when you hear00:02:48
this is the definition of why I am usually a student00:02:51
I will tell this topic in a slightly different way00:02:53
I will tell you such a story00:02:55
hope very soon it will happen in00:02:57
in real life, the only thing I have is the mouth00:03:00
make remarks I usually say that00:03:01
Kristina nas Sorry if you were crying courses00:03:04
if there are many things in one car, we are not00:03:07
let's get a seat, we need to order a bus00:03:09
Let's pretend I'm not a car there yet00:03:11
there is a bus. Yes, you and I all sit down and00:03:14
we are going to Kherson to rest on the sea00:03:16
see what roads there are, there are some. These are the ones00:03:20
So there are some circular ones with some00:03:23
we are so expensive with such dwarfs00:03:25
we don't take it. This is not suitable for us00:03:27
why should we go like this?00:03:29
you and I choose crooked roads00:03:31
some such and such a flat road so that00:03:34
direct Yes, but in any case00:03:36
even This is the road that goes in principle00:03:38
in a straight line without various dwarfs there and00:03:41
curl This road has pits and this00:03:45
the road has some hills00:03:47
localities so That is, Well, I she roads in00:03:50
bumps in the roads are very popular with us00:03:52
also if well relevant because if we00:03:54
we are talking about Western Ukraine. There it is00:03:56
quite such a widespread phenomenon I00:03:59
if we are with you now, this is the road00:04:01
let's transfer it to the starting point. This is the road -00:04:05
this is a function and the derivative between you and me00:04:07
helps to understand where functions are00:04:10
where these bumps are located and where in the fons00:04:13
these pits are located here and it helps00:04:16
let us estimate how deep the pits are and00:04:19
how tall is your hump, that is00:04:21
the function is our road and the derivative is this00:04:24
our tool that helps us00:04:26
evaluate what kind of road and what kind of function00:04:30
some dimples and where are you now00:04:33
figured out what a derivative is. If so00:04:36
I'm drowning a plus in the chat, if not, then a minus in00:04:38
chat and questions that are not clear. It is clear00:04:41
yes Dear, we are all getting on the bus00:04:43
let's all go to Kherson and leave for us00:04:46
helps to understand where the pothole is on the road A00:04:49
where there is a derivative in the hill on the road00:04:51
characterizes the rate of change of the function00:04:54
How quickly this pit is formed00:04:57
because quickly therefore on deep so what00:04:59
she deep shows us How much00:05:01
that and how much He is formed quickly00:05:04
high i.e. the derivative characterizes00:05:06
rate of change function How fast00:05:09
our path is changing, this is for you and me00:05:12
helps to understand the derivative process itself00:05:15
finding the derivative is called00:05:18
in a smart word differentiation that is00:05:20
when I'm looking for a derivative I do an operation00:05:23
which is called00:05:25
differentiation in order to seek00:05:28
You just have to know the derivative00:05:30
use the whole tablet again00:05:33
I emphasize that you will have it in00:05:35
reference materials are not needed00:05:36
to worry that something is there if Well you are not00:05:39
you have to study or cram because I used to00:05:41
I crammed her time because I wasn't there for her00:05:44
ZNO is now there, so don't worry00:05:46
all you need to know is just to know it00:05:49
use everything. That's how it is00:05:51
looks I want now from this plate00:05:55
select the main ones and the most popular ones00:05:59
derivative functions that are there most often00:06:01
used to tell you a little about them00:06:03
explain a little about them tell on00:06:06
I will tell you about the next slide now00:06:08
about such basic derived functions first00:06:12
for all the derivative of a number, see00:06:15
plates and at the external examination and, in principle, everyone has them00:06:17
plates that you will see there in00:06:19
You will google on the Internet and write Here00:06:21
thus it is a derivative derivative00:06:24
is denoted by such a cut, here is this one00:06:26
a dash everywhere means that the derivative is taken00:06:29
the derivative is zero, so what is it?00:06:32
tsechka is from the word constant If you00:06:35
you know English well, you understand what00:06:37
constant is something constant something constant00:06:40
a constant is some number, i.e. this00:06:43
derivative is zero, this means that00:06:46
the derivative of a specific number00:06:48
is equal to zero, for example, the derivative of four00:06:51
four derivative is zero further00:06:55
the derivative -3 is also a constant, also a number00:06:59
derivative -3 is also equal to zero and so on00:07:03
and then the derivative of absolutely any00:07:06
number is always equal to zero, it is a derivative00:07:09
is equal to zero cell once again00:07:11
the constant is emphasized, this notation00:07:13
some specific number further00:07:16
the derivative of a variable is the derivative of simply x of the variable00:07:21
is equal to one, it is easy in principle00:07:23
derivative x = 1 but the derivative is very important00:07:28
which is very often used here in00:07:31
on the tablet it is written in us and you in00:07:33
the third line is the derivative of x in00:07:36
degrees, in principle, the formula here is enough00:07:39
It's still well written, but I'm not here for you now00:07:41
I will tell you an example again and tell you00:07:44
more specifically, how to find the derivative of x in00:07:47
degrees, for example, let there be x³ well00:07:50
for example, yes, the first number is for me00:07:52
it occurred to me that x^3 should be found00:07:55
derivative look carefully now at the formula00:07:59
what is being done here00:08:00
the x indicator climbs forward as00:08:04
the multiplier of porridge became the multiplier ahead00:08:07
this is the x indicator for me x^3, respectively00:08:11
the triple climbs forward as the multiplier will be00:08:14
3 multiply by and What happens next00:08:18
the indicator x will decrease by one was00:08:22
x^300:08:23
will become x^2 I found the derivative of the derivative of x00:08:28
to the third power is equal to 3 *00:08:31
x in the second degree, the indicator climbs out00:08:35
forward as a factor and exponent00:08:38
everything decreases by one. That's it00:08:41
in this way, the derivative x in some00:08:43
there powers all other derivatives here00:08:45
it's obvious from the sign it's simple00:08:48
the derivative of the sine is the cosine derivative00:08:51
cosine - sine has nothing to do with it00:08:52
to speak and what to teach and opened the tablet00:08:55
I looked and wrote down the only thing that is here00:08:58
unique is the derivative x in some00:09:00
degree because degree can every time00:09:02
be different can be there third fifth00:09:05
there 105 This is how you should understand the algorithm00:09:09
by which this indicator works in advance as00:09:12
multiplier and show that we reduce by00:09:14
unit was x^3 became x^2 we return00:09:19
here the derivative is simply x = 1 and in fact it is00:09:23
logically because if in principle00:09:25
apply for simple x here is this one00:09:28
look at the formula, it will also work00:09:30
derivative X derivative x simply x - That's what00:09:35
such This is X to the first degree true here00:09:38
the unit on the drag works in the same way00:09:41
the multiplier will be one multiplied by and sign00:09:44
let's reduce the indicator by one, it was x c00:09:47
reduce by one to the first power00:09:49
will be X to the zero power of X zero00:09:53
degrees This is what is This is a unit unit00:09:56
I multiply by one, I get one, that is00:09:58
in principle, it is logical that the derivative x =00:10:01
1 is quite obvious from this formula00:10:05
which I just showed you, you figured it out00:10:07
the derivative of a constant of some number is equal to00:10:10
the derivative of x = 1 and the derivative of x in are zero00:10:16
I showed you how to find degrees and more00:10:19
hard to hear one moment and another00:10:21
normal to hear00:10:24
everything is fine, it's still special00:10:27
derivative there are a lot of memchiks and00:10:30
I even inserted one of them here00:10:32
slide derivative exponent in power00:10:35
is always the same if I take the derivative00:10:38
from the exponent of the power of x it will be so00:10:42
what is the exponent to the x power00:10:44
the exponent of the exponent is this number00:10:47
a mathematical constant just like a number00:10:49
Pi We have Similarly, the exponent If00:10:52
the number Pi is approximately 3.14 there00:10:56
exponent is approximately equal to00:11:00
2.72 is also just a number that is denoted00:11:04
letter and this is a mathematical constant and that's it00:11:07
this is a good idea for you so that you feel better about it00:11:09
fixed and I update update update00:11:12
exponent and it remains by itself00:11:15
how many times because I did not take the derivative of00:11:17
exponent She will always be herself00:11:19
everything you need to know about derivatives is there00:11:23
some sun I will now tell you a little about them00:11:25
I will tell you in more detail. But your biggest00:11:28
a friend in the derivative topic is this plate00:11:31
sign and let's go, let's go on to here00:11:35
of these nuances, the first is a derivative of Suma I00:11:38
differences if I have several functions00:11:40
so they are added to each other or00:11:43
are subtracted How should I search00:11:45
derivative if I have several functions00:11:48
which add or subtract to each other00:11:51
I'm just looking for the derivative of each separately00:11:54
functions and if there was a plus THEN I add How00:11:57
between themselves, if there was a minus, then it subtracts00:12:00
give them to each other00:12:02
let's practice now with you, let's look at00:12:04
these derivatives and so the first function y =00:12:08
4x + 6 we are looking for the derivative y, the stroke is equal00:12:13
See if before your function with00:12:16
absolutely any tablet stands00:12:20
some coefficient, for example, there is four00:12:23
If you remember X, then you are it00:12:26
coefficient simply do not touch the same as00:12:28
now we have four standing here00:12:30
multiply by X the coefficient before x00:12:33
it is worth multiplying by four this coefficient00:12:36
coefficient you just Do not touch so and00:12:40
you leave if there were 4, then I don't have a four00:12:44
I touch like this and let four be multiplied by00:12:47
I'm dealing with x derivative of x Yes, yes00:12:52
and write x' x' like this00:12:56
x derivative Here is X3 + derivative00:13:00
the derivative of the following second function is 6 so00:13:04
I am writing to you 6 stroke is equal to 4 as00:13:09
we leave the coefficient untouched00:13:12
derivative x I look at the derivative plate00:13:16
x is equal to one, that's how I write it00:13:19
multiply by 1 + 6 what is the derivative equal to00:13:24
6 is a constant, it is some number I00:13:27
I look at the plate and see that it is derivative00:13:30
constant, the derivative of the number is zero00:13:32
accordingly, the derivative of 6 is all zero00:13:36
is equal to 4 * 1 = 4 + 0, so it will be 4 in00:13:44
in this case it turned out to be so unique00:13:46
the case when the derivative is equal to the number no00:13:49
the derivative must always be equal to a number00:13:51
It can equal someone's expression00:13:54
alphabetic, but maybe a number like that00:13:56
a unique example where the derivative is equal to00:13:58
the specific number 4 can also be the case with00:14:01
this example has cleared up whether there is a question00:14:04
I will explain why there are four left00:14:07
if before your table function00:14:09
there is some multiplier, coefficient, number00:14:13
so we don't touch that number00:14:16
we just leave the coefficients like that00:14:18
remain, we do not take the derivative of00:14:21
See also00:14:24
for example, I want to show if here00:14:26
will be 4 * Sin x then the quadruple coefficient i00:14:30
I don't touch it, I take the derivative only of the sine of x00:14:33
here the same predexom stands00:14:36
factor 4 factor We leave it00:14:40
number if your table function00:14:42
is multiplied by a number simply by one00:14:45
a number is a coefficient, you don't touch it00:14:48
looking for the derivative of a function only00:14:50
figured out the following example y =00:14:54
6x² - the exponent to the power of X is gone00:14:58
derivative y dash = and so we look at 6 *00:15:04
in the second degree, the first six are this00:15:07
coefficient before my table function00:15:11
x^ So x² is a function of the x power00:15:15
the tabular coefficient is 6 6 I do not00:15:18
I touch it like this and leave it. Yes, because this is the coefficient a00:15:22
I take the derivative of x to the second power of x00:15:25
I take the square derivative minus the beru00:15:28
I take the derivative of the second function00:15:31
exponents to the power of X exponents00:15:34
powers of X =00:15:36
6 I'm going to fill in such ringworm, we're going to deal with it00:15:41
the square derivative of x today00:15:44
they already talked about it derivative x in00:15:47
power indicator climbs forward as00:15:49
the multiplier will be 2*00:15:54
decreases by one was x in the second00:15:58
powers reduced by one will become x c00:16:01
I forgot the minus of the first degree here00:16:04
draw a dash to work minus00:16:06
we are looking for the derivative of the exponent in the power of x00:16:08
it is this unique function so less about00:16:12
which I showed you today as a derivative00:16:14
exponents are always exponents, that is, and00:16:17
there will be an exponent to the power of x a little bit i00:16:20
now let's tame it all, let's give it six00:16:23
multiply by 2 will be 12 x to the first power00:16:27
Well, just x and write like 12x -00:16:31
exponent to the power of X I found everything00:16:34
the derivative of the difference of two functions, i.e. i00:16:37
I am looking for the derivative of each function separately and00:16:40
then I add or take them away from each other00:16:42
I figured it out. If so, please00:16:45
pluses, if not, then the question What not00:16:48
Of course, it's great, then we train for it00:16:51
examples from the ZNO ZNO of the 17th year are necessary00:16:55
find the derivative of the function y = sin x - cos x00:16:59
+ 1 is the derivative of the Sum of the sum and the difference here00:17:03
even such a little bit, there is a little bit of difference00:17:05
There are sums, but we work according to the I take scheme00:17:08
separately the derivative of each function00:17:11
fund it is and accordingly subtract them between00:17:14
myself or add a y stroke00:17:17
is equal to the first that they stand sine X Y00:17:21
I am looking for the derivative of sine x table00:17:24
I open it and look at the derivative of the sine x =00:17:29
cos x All please I come back here and00:17:33
I write cosine x cosine x stands for minus i00:17:38
I write - then the derivative of cosine X is whatever00:17:43
laska I open the tablet and look00:17:45
derivative of cosine X = - Sin x is good00:17:51
I return and write the derivative tax - Sin x00:17:55
pay attention now to these disadvantages00:17:58
the first minus is this minus I have with00:18:01
conditions because between the functions In the condition it is I00:18:05
in the minus I also write a minus and here is this one00:18:08
the second minus from the derivative because it is a derivative00:18:11
cosine equals minus sine x here00:18:15
It is important not to get confused with these00:18:17
with minuses, then the derivative of units of units00:18:21
- is What is a constant, is a number, and is a derivative00:18:24
any number equal to zero has a plus00:18:27
0 Let's figure it out now here minus on00:18:31
minus gives me plus So I have cosine00:18:35
plus sin x cos x + Sin x Well, this one + 000:18:41
there's no point in dragging me around anymore00:18:43
+ 0 t + 0 all cosine x + Sin x variant00:18:47
I earned D + B on the external examination, which you did not00:18:51
clear in this task if00:18:53
the coefficient stands in front of the cosine we00:18:55
we will also leave it like that and not touch it00:18:57
there four multiply by cosine x father00:19:00
four, I'll leave it like that and don't touch it00:19:02
that is, if you have this table in front of you00:19:05
function stands for the coefficient you that00:19:07
the coefficient is left and taken00:19:10
the derivative of your tabular function is all00:19:14
it's so easy, no one said what would happen00:19:16
we look forward to the next one with you00:19:18
the example is also from ZNO, see many00:19:22
here you can think that it is so good here00:19:26
standing multiplication must be used00:19:28
the formula for the derivative product is not here00:19:33
you just need to open the brackets. Let's do it00:19:36
now with you before looking for a derivative00:19:38
to begin with, we will open the brackets a little00:19:41
let's simplify that function00:19:43
open the brackets x multiply by x³ will be00:19:49
x^4 then we multiply x by plus one will be00:19:54
plus X and now you clearly see what we have00:19:58
ordinary derivative Sums of two functions all00:20:01
no more. We have opened this product00:20:04
brackets and got much simpler00:20:07
function and we have a lot of it now00:20:09
it is easier to find the derivative, we are looking for the derivative00:20:12
Sums of two functions00:20:14
separately derivative of this function separately00:20:17
I add the derivative of this function and00:20:20
I'm writing down the answer, let's go F dash00:20:23
derivative F stroke derivative is equal to left00:20:26
went and so00:20:28
derivative of x in the fourth power of x in00:20:33
to the power of 4 in advance as a factor of four00:20:36
multiply by the sign of the indicator, reduce by00:20:39
unit was x^400:20:43
will become x³ + we move on derivative X =00:20:49
why is the tabular derivative of the tabular clearly00:20:52
it is written that the derivative x = 1 Well, that's all00:20:55
please 4x³ + 1 option a plus B00:21:00
did What is not clear No bunny x to00:21:03
X is not added, we are with you. And you have00:21:05
I mean, no, because it's not00:21:07
similar terms are completely different in them00:21:09
indicators here X1 here x^4 we cannot with00:21:14
add both because these are not similar terms00:21:16
absolutely two different expressions so we them00:21:19
we leave it like that00:21:21
where did one look, we multiplied x00:21:26
by 1 we got + x That is, we simply00:21:29
they opened the brackets 1 disappeared someone asked00:21:32
what four in advance is the formula according to which00:21:34
we are looking for the derivative of x in power00:21:36
the indicator climbs forward as a factor a00:21:39
the sign and pointer will decrease by one00:21:42
four on before as a factor and an exponent00:21:45
reduced by one was x^4 became x³00:21:49
all went further, the next example too00:21:53
from ZNO, we deal with a derivative like this00:21:56
functions function is so complex on00:21:58
first look but actually let's00:22:00
let's figure it out, you and I have one function00:22:03
the second function and the third one are added between00:22:07
I am looking for it now00:22:09
the derivative of each function is given below00:22:12
I add and subtract between myself and everything is fine00:22:15
all formulas in the table of derivatives will be obtained00:22:17
there is no such thing as this Super A that is important00:22:20
not super, all the formulas there are important00:22:22
remember and understand Well, how to understand00:22:25
remember you they will be on additional00:22:27
yes, but all of them are important00:22:29
everyone is asked with the same frequency00:22:32
which everyone will ask about at the final exam00:22:34
asked all met all important00:22:37
let's go and look for the derivative of the game is equal to00:22:41
first of all minus 7/6 what is it00:22:45
the coefficient that I do not touch is 7/600:22:48
the number is a multiplier, I leave it as it is00:22:51
minus 7/6* then the derivative x in the sixth00:22:56
to the power of 6 indicators in advance as a factor00:22:59
decrease the sign of the indicator by one00:23:03
x^6 is the state of X5 to the power of all plus 5 too00:23:09
I don't touch the coefficient with a five00:23:12
I am looking for the derivative of x in the fourth00:23:16
to the power of 4, the multiplier climbs out00:23:19
decreases the exponent sign00:23:21
indicator by one will reduce the indicator00:23:24
per unit was x^400:23:26
will become x^3 - derivative 14 14 is a constant00:23:33
this number is the derivative of a constant number00:23:36
is equal to zero, let's cultivate a little. That's all00:23:39
and everything will be super six six00:23:41
decreases, remains -7 -7 * x^5 What00:23:49
here 5 * 4 will be 20, we have plus 2000:23:54
x^3 and this -0 makes no sense to write 0,000:23:59
everything is minus and 7x^5 + 20x3 is mine00:24:05
the answer is the option where it stands here00:24:06
option in so so wonderful Everything is a plus point on00:24:10
ZNO did What is not clear here00:24:14
how to understand where to put a dash What are you?00:24:17
you mean a bunny where to put a dash00:24:19
You should put a dash next to the fact00:24:22
of each of its own particles Near each of its own00:24:25
the term should be crossed out, that's all00:24:28
so00:24:33
Okay, we figured out the next example too00:24:36
ZNO needs to find the derivative of the function we went00:24:39
y dash = everything according to the y dash standard00:24:43
is equal to the fourth power we are looking for00:24:46
derivative 4 in advance as a multiplier text 4x300:24:50
the multiplier and the sign of the exponent are reduced by00:24:52
unit was x^4 becomes x³ + what here here00:25:00
the factor of three is multiplied by the cosine00:25:03
x factor 3 I don't touch it like that and leave a00:25:07
I am looking for the derivative of cos x the derivative of cos x What is there00:25:11
I need a plate. Yes, a plate00:25:13
we look at the derivative cosine x = - Sin x00:25:18
great, I'm coming back here00:25:21
I come back and write - sinx - Sin x A little00:25:28
I multiply, simplify, and everything is super00:25:31
come out 4 hobbies and it will be00:25:35
4x³. and 3 multiplied by - Sin x will be -3 Sin x00:25:41
- 3 Sin x yes yes -3 sinx all i00:25:45
I boldly answer option b00:25:49
no, no, in, that's how it is here without the third00:25:54
degree option in and everything and I earn a point00:25:57
it was not difficult at the external examination, you take it00:25:59
add the derivative to each piece separately00:26:02
or subtract and everything turns out great00:26:05
excellent further further derivative in points very00:26:08
they often ask students to calculate the derivative of00:26:12
students are afraid of a specific point and00:26:14
are afraid, but in fact it is elementary00:26:16
see For example, I have a function Here00:26:19
so it is necessary to find the derivative at the point x000:26:23
= 6 x zero This is how it is denoted00:26:26
some specific point, the coefficients x000:26:28
or just x don't get scared just like that00:26:30
all indicate a specific point00:26:33
nothing else as I am looking for the point derivative00:26:37
I'm simply looking for the derivative first00:26:40
and then I put a number instead of X in the derivative00:26:44
6 and count a specific number We are looking for everything00:26:48
derivative and00:26:51
y'= I am looking for the derivative of such a function00:26:55
four is a coefficient, I don't touch it00:26:58
the derivative of x to the second power is two00:27:02
forward as a power factor and reduces by00:27:06
unit was x^200:27:08
will become X1 + derivative 600:27:14
- this is a specific number, this is a constant00:27:17
the derivative of the constant is zero00:27:20
we calculate 4 * 2 will be 8 and X in the first Well00:27:24
so there will be X and plus 0, there is no point in writing00:27:28
I always asked for the derivative in point 6, what do I do00:27:32
Instead of x in the derivative, I put the number 600:27:35
and count a specific number All I multiply by 800:27:40
6 I get 48 derivatives of such a function in00:27:44
we are not afraid of the point x0 = 6 = 4800:27:49
this type of question found a derivative00:27:52
substituted a number in the derivative instead of x00:27:55
which they ask for and found nothing of the sort00:27:58
there is no supersky here, where does zero come from?00:28:01
the derivative of the hexadecimal bunny 6 is a number00:28:04
is a constant, the derivative of a constant is equal to00:28:07
zero, that's where our zero came from00:28:12
Okay, here we figured out X zero00:28:15
we do not pay attention in any way Well x0 x000:28:18
all that is, it's just a notation like this00:28:21
we all train for a specific point00:28:23
now with you, for example, from ZNO is necessary00:28:26
find the value of the derivative like this00:28:28
function at the point x0 = -1, we look for the F line00:28:34
from x is equal to the first thing we start with00:28:37
do this Find the derivative of such a function00:28:40
dachshund two is a coefficient I do not touch00:28:44
such lichens we are looking for the derivative of x in00:28:47
to the third power of three is an indicator00:28:48
the multiplier a climbs out00:28:52
indicator decreases by one unit00:28:54
x to the third power will become x to the second00:28:58
powers minus the derivative of fives00:29:01
five is a number, it is a derivative constant00:29:05
number is zero, we calculate everything 2 * 300:29:09
there will be 6, there will be 6x squares Well, -0 and -0 are all00:29:14
I asked at the point x0 = -1 perfectly00:29:20
instead of X, I put Minus one and count00:29:23
the value of the derivative point x0 = 100:29:26
we substitute 6* instead of X I put -100:29:32
Yes00:29:33
squared is equal to00:29:36
-1² This what is This one I have 6 * 1 = 600:29:43
all my answer option d + score on the external examination00:29:47
I earned literally there for Well, I don't know00:29:50
to do this task in a minute00:29:53
have they really figured it out at all?00:29:55
question on this00:29:57
When you can write that F stroke well if00:30:02
so correctly F stroke from minus one00:30:06
is equal to so so correctly write down ot00:30:09
But yes, you said everything correctly if I00:30:12
I correctly put a single instead of X00:30:15
I can write that the F stroke instead of X is 100:30:18
from -1 = and I put -1 but this is a test00:30:22
format and how you solved it00:30:25
no one will question your importance00:30:27
was the correct answer, everything is okay00:30:30
let's continue to be more interesting with such notes00:30:34
Trigonometry has to be found00:30:36
the value of the derived point x0 = pi / 2 Well00:30:41
we know the principle of looking for the derivative first00:30:44
for everything, then instead of X I put pi / 2 and00:30:47
I'm counting. Let's go F00:30:50
the line from x F the line from x is equal00:30:54
in front of the cosine is the quadruple Tse00:30:57
just the ratio accordingly 4 I don't00:31:00
I touch this coefficient and leave it at that00:31:02
multiply by the cosine derivative opens00:31:07
tablet Today we will go to Kos with you00:31:09
have already seen the derivative there 150 times00:31:11
cosine equals -7 I write - sinx - Sin00:31:19
x + the derivative of the five-point derivative of the constant00:31:24
of a specific number is zero00:31:27
let's cultivate we went 4 multiply by - Sin00:31:31
x will be -4 Sin x Sin x and plus 0 we00:31:37
let's write It's obvious that everyone asked to find it00:31:40
give the derivative at points pi / 2 perfectly00:31:43
let's correctly write F stroke instead of X00:31:47
I put pi / 2, so F is a stroke from pi / 200:31:51
= I just instead of x in the function here00:31:55
I put pi at 2 and count what I will get00:31:58
will be minus 4 multiplied by the sine of pi / 200:32:03
on the sine of pi / 2 is great the sine of π / 2 is00:32:09
same as sine00:32:11
90°. it is the same as the sine of 90°. By the way00:32:15
the plate that is given to you at the ZNO post office at00:32:18
nmt there, angles are also written in radians pi /00:32:22
2 and in degrees 90 Well, in principle, without00:32:25
difference plate opened sinuses00:32:27
cosines and We saw that sin π / 2 = 100:32:32
I have minus four multiplied by 1 will be00:32:36
-4 all option a + point earned difficult00:32:41
not difficult, well, I hope that they are difficult00:32:44
the sign will be 100%, us00:32:47
if last year they gave it this year00:32:49
will also give I think in general if it is already00:32:52
not even discussed like that00:32:53
with the derivatives of the sum and the difference of two functions00:32:57
we figured out if is more interesting if00:33:00
the derivative of the product of two functions or00:33:03
shares are here Unfortunately or fortunately That's how it is00:33:06
iridescent doesn't work here, you can't take it00:33:09
separate derivative in the first function separately00:33:12
derivative of the second and transfer them between themselves00:33:15
it is very important not to bring Bozhechko00:33:18
In this way, not to be mistaken, because very often00:33:20
students find the first derivative00:33:23
functions found the second derivative00:33:25
multiplied and everything turned out great in them no00:33:28
with the derivative of the derivative with the derivative of the share00:33:32
these special formulas work00:33:35
the formulas immediately say that the slide has not changed00:33:38
everything else has changed for me00:33:42
your slide has changed changed means in00:33:45
who hasn't changed, then reset the 14th roof00:33:47
everything is okay and there should be 11 likes from00:33:50
the initial product of the derived quotient00:33:52
these special formulas work00:33:55
You will have formulas in additional ones00:33:58
materials, so don't worry about it anymore00:34:00
what if you don't need to serrate them directly00:34:03
because they will be at your fingertips, but you00:34:06
you must clearly understand how they00:34:08
went to use derivative products00:34:11
if I have two functions they are marked00:34:13
as a function in This is the first function and a function00:34:16
In this second function, the formula looks like this00:34:19
thus first I take the derivative00:34:23
the first function and simply multiply by the second00:34:27
function plus simply take the first function and00:34:32
multiply by the derivative of the second function, i.e. i00:34:35
I take turns00:34:37
derivative of the first and simply the second and vice versa00:34:41
simply the first and the derivative of the second alternately00:34:45
I take the derivative of the first function, then of the second00:34:48
and let's add these products together00:34:51
let's practice multiplying x squares by00:34:55
exponent to the x power Here is my first one00:34:59
function Here's my y Here's my second function00:35:03
Here is mine in let's try the formula00:35:08
Oh stroke I first take the derivative first00:35:11
functions I take the derivative of x squared00:35:15
so x is squared and I just multiply by00:35:19
I simply multiply the second function by V. I simply00:35:22
I multiply by the exponent to the power of x + now00:35:27
on the contrary, I do not touch the first function of X00:35:32
I do not touch the squares and take the second derivative00:35:36
function, I take the derivative of the vertex in the given00:35:40
case, I take the derivative00:35:43
exponents to the power of x again I first00:35:47
took the derivative of the first function and multiplied it00:35:52
just for the second plus just took the first00:35:56
function and multiplied by the second derivative00:36:00
the derivative of the exponent to the power of x is now this00:36:04
I count everything and everything should turn out great, let's go00:36:07
x dash derivative is equal to x² derivative00:36:11
is equal to the two in front as a factor00:36:14
and the power of X decreases by one was00:36:18
x in the second power will become x in the first00:36:22
power and multiply by the exponent in00:36:26
to the power of x plus how X stood in the second00:36:30
powers and stands and the derivative00:36:33
exponents to the power of x will be so00:36:36
the exponent of the power of x is equal to that I can00:36:40
do x to the first power and X will be00:36:43
I can simply write how to multiply by 2x00:36:47
exponent to the power of x and plus x²00:36:51
multiply by the exponent to the x power00:36:55
alternately take the derivative of the first function00:36:59
simply to the second plus vice versa simply00:37:02
the first to the derivative of the second. This is the formula00:37:06
you will have extras on hand for the external examination00:37:09
materials so don't panic there she is00:37:11
it will but you have to understand how it works00:37:14
What is it, what is it for V And in general, how is it00:37:17
use Now let's share00:37:20
on the examples of ZNO now with you00:37:22
let's practice, don't worry, let's go00:37:24
by share00:37:25
it is more difficult to share, that is, to divide taxes00:37:29
of two functions that 2x because x to the second00:37:32
power derivative x in the second power00:37:34
will be 2x so the two in advance as a factor and00:37:38
the power of x decreases by one will be X00:37:40
in the first power and x in the first power00:37:43
it's just x with a fraction here that's more complicated00:37:47
let's figure out how to remove x by brackets00:37:49
you can and exponent you can What is the difference00:37:52
i.e. add another board here00:37:55
Option It is possible, but I am in this sense. Honestly00:37:57
saying not there, I don't really see, let's go00:37:59
then the quotient derivative of the function U is divided00:38:03
to the function V00:38:05
the dial looks exactly the same00:38:09
in the same way as the derivative of the product only by00:38:13
in the middle there is a minus in the same way00:38:15
alternately you take the derivative of the derivative00:38:18
simply multiply the numerator by the denominator00:38:22
simply multiply the minus by the denominator00:38:27
on the numerator, that is, on the contrary, everything is tax00:38:30
the numerator of the derivative is simply multiplied by00:38:34
the denominator minus just the numerator00:38:37
is multiplied by the derivative of the denominator in00:38:41
the principle of the formula once again I say identically00:38:43
to the product but in the product in the middle00:38:46
there is a plus a in parts along the middle00:38:50
there is a minus and that is not a big life hack with00:38:53
which I always tell my students when you see them00:38:55
derivative of the fraction The first thing you do is this00:38:59
write the fraction line and raise the denominator00:39:03
to the square This is where you should start00:39:07
find the derivative of a fraction, write a dash of a fraction00:39:11
to seek to raise immediately in the denominator to00:39:14
square and then figure it out00:39:17
numerator with numerator everything works as00:39:20
with the derivative product but in the middle00:39:23
there is a minus when 4 is divided by the cosine00:39:28
x and the derivative, the first thing I do is I write00:39:32
I bring the fraction line and the denominator to00:39:35
square denominator cosine x I bring00:39:39
I write cosine x² to the square. everything is cosine00:39:43
square X is great now we understand00:39:46
numerator, first of all, in form00:39:49
derivative of the numerator I have a 4x derivative00:39:54
derivative of the numerator00:39:56
is simply multiplied by the denominator is multiplied00:39:59
simply by cosine x by cos x - simple00:40:05
the numerator is just the numerator, it's them00:40:08
the derivative is simply the numerator 4x multiplied by00:40:13
the derivative of the denominator cosine x by the derivative00:40:17
denominator00:40:18
cosine x is all alternate derivatives00:40:22
numerator to denominator00:40:25
refrigerator to the derivative of the denominator00:40:26
equals, now we will calculate the derivative 4x =00:40:31
I just don't touch the 4 coefficient, but the derivative00:40:34
how will everything be, the unit has four00:40:36
cosine x four cosine x -00:40:41
Multiply by 4x according to the derivative table00:40:46
cosine equals -7, so I write it - Sin00:40:51
x - Sin x - x and divide by cosine00:40:58
squared cosine squared X is great a little bit00:41:02
we transform, simplify the numerator and everything in00:41:05
you and I make a great four00:41:07
cosine x is I don't touch it and leave four00:41:11
cosine x here00:41:13
-4x I multiply by - Sin x will be + so therefore00:41:17
that minus on minus will give me a plus has00:41:19
plus four X multiply by sine x by00:41:25
son and divide all this happiness by the denominator00:41:28
cosine square of X is simple here00:41:31
transformation Frankly speaking, already gone00:41:33
it is important to understand at the very beginning00:41:36
derivative of What you should take immediately What00:41:40
you dash the denominator of the fraction to00:41:43
of the square and with the numerator everything works as with00:41:47
derivative of the product, but in the middle you00:41:50
put minus00:41:52
Yes, let's see what is not clear about this00:41:54
there will be a classic question at the very end00:41:57
details that I am not soon00:41:59
next great great great next Well00:42:03
let's train on examples from ZNO, let's go00:42:06
let's see here that y = x in this power00:42:10
I will not multiply the slides back00:42:15
a second back for a second let's go back00:42:17
someone there just turn on the 13th slide I already am00:42:20
I will not return, you will have a record of you00:42:21
you can turn on the recordings later00:42:23
pause overwrite record00:42:26
for screenshots That is, It's not a problem Okay00:42:29
derivative of the product Well, let's use the formula Я00:42:33
I am looking for the first derivative of the first function00:42:37
my function is X to the seventh power00:42:40
so X is the derivative of the product to the 7th power00:42:46
I simply multiply by the second function00:42:49
lnx is not intu LN but it is the logarithm of00:42:53
the base of the exponent is the logarithm with the base00:42:57
exponent everything just has something special00:43:00
Elen's designation is natural00:43:02
logarithm further plus now the first function x00:43:07
in the seventh I do not touch but look for a derivative00:43:11
the second function00:43:12
I am looking for the derivative of Elen x so with the derivative00:43:17
the second function and now we understand00:43:20
x-derivative to the 7th power What is this00:43:23
seven in advance as a multiplier will be 700:43:25
multiply by and the sign of the exponent decreases by00:43:28
the unit was x in this state x^600:43:33
multiply by lnx multiply by lnx00:43:38
multiply by lnx + X in the seventh X c00:43:45
at seven and look carefully at the sign00:43:47
what is the derivative of LX i to you00:43:51
I specially open the sign so that you00:43:52
they saw it with their own eyes. Here it is00:43:55
please don't bother with everything on the plate00:43:57
derivative LN = 1 / x All please Ya00:44:04
I return to myself00:44:06
that's how I lost him. That's how I'm coming back00:44:09
I write that the derivative lnx = 100:44:14
/ x = is equal to something else can be done here00:44:18
with the first preposition Nothing because I his Yes00:44:22
and I leave 7x in the sixth on lnx and here is z00:44:26
something can be done like this with the second term00:44:29
that today Ellen is somehow very crooked00:44:31
Alex is a dachshund, and with the second addition, you can00:44:35
work a little and reduce 1x and 1x00:44:40
accordingly, if there was x in this power00:44:43
will become x^6 i.e. X in the denominator with this00:44:47
x, I shortened this one by 1X a little00:44:50
the indicator decreased, everything became X in the sixth00:44:54
powers why on 7x^6 That's how I have it00:44:57
bunny 7x^6 That's it, everything is fine00:45:00
still found the derivative x in the seventh00:45:02
- I still have this 7x^6 right here00:45:06
I liked standing multiplication by lnx, I do00:45:09
and I write multiply by l x all mine00:45:12
the answer is option b and everything is fine00:45:15
it turns out why not x^7 And what should be X700:45:19
When I look for the derivative, I reduce it00:45:22
indicator per unit if there was x in it00:45:25
then it becomes x^6 That is, it cannot be there00:45:29
a priori X800:45:32
What is not yet clear about this task00:45:34
there is still a question00:45:36
why plus x and the formula see the formula00:45:41
there is a plus here plus and I write plus plus00:45:45
everything according to the formula, the formula plus stands between00:45:49
two pieces and I write a plus between the two00:45:52
in pieces. OK, let's give with a share00:45:55
find the derivative of this function00:45:59
derivative of the quotient two functions are divided between00:46:02
what I told you when you00:46:04
see the derivative of the fraction The first thing you do00:46:08
in general, you write the dash of the fraction and00:46:12
to bring the denominator up to the square in me00:46:16
the denominator is x the first time I do it00:46:19
carry the denominator to the Square of x00:46:21
I bring you to the Square once again00:46:24
I will show it on the Formula look carefully at00:46:27
The first thing we do is to derive the formula00:46:30
we bring the denominator to the fraction00:46:33
of the square there I make Similarly I from x00:46:37
I bring the denominator to the square, or not00:46:40
visible00:46:43
which is not visible00:46:44
letters near00:46:47
everything is visible00:46:48
someone that is not visible00:46:51
everything should be visible everything is visible perfectly then00:46:54
we went further, we work in the same way00:46:58
as with the derivative of the product I start with that00:47:01
what I'm looking for, I'm looking for the derivative of the numerator00:47:05
I write down the derivative of the numerator for you00:47:08
2x - 3 derivative of the numerator00:47:12
I simply multiply it by the denominator00:47:15
simply by X then minus and vice versa simply00:47:20
numerator 2x - 3 2x - 300:47:26
multiply by the derivative of the denominator multiply by X00:47:31
derivative, now I am looking for derivatives, converting00:47:35
expression and everything works out great for me00:47:37
went F dash derivative00:47:40
F stroke derivative is equal to let00:47:43
deal with this 2x derivative what is this00:47:48
simply 2 coefficient of two Does not touch a00:47:51
derivative x = 1 - 3 is a constant derivative00:47:58
number, the derivative of the constant is zero00:48:01
write plus 0 There's still no point00:48:04
it is obvious that the derivative of 2x = 2 derivatives00:48:08
of threes is equal to zero, all derivatives of this one00:48:12
of the entire parenthesis of the entire expression is equal00:48:15
the prime number 2 is all the prime number 2 and00:48:19
multiply by X, respectively, have 2x -00:48:24
let's figure it out here00:48:26
derivative x = 1 if I multiply the expression by one00:48:31
then nothing changes, it will be the same00:48:34
I write the expression 2x - 3 because it is a derivative00:48:38
x = 1 if I multiply the expression by one then if00:48:43
something doesn't change very much there00:48:45
divide by x² divide by X squared00:48:49
now you need to open the parentheses Before which00:48:52
stands on the minus as I do it parentheses00:48:55
minus I clean And all signs are in brackets00:48:58
I change the minus brackets to opposite ones00:49:00
removed the signs in parentheses, changes to00:49:03
opposite stood I have the positive 2x00:49:08
will become negative, accordingly, it will be -2x00:49:11
stood I have a negative -3 should become00:49:16
positive + 3 = 2x and -2x cancel each other out00:49:22
all that remains is to divide by three00:49:26
x². all option a + score on the external examination is ready00:49:31
everything was done00:49:35
without cartridge00:49:37
my bead00:49:39
what is not visible to whom what is not visible00:49:45
no answers as no answers00:49:48
If you are from the phone, then it is possible00:49:51
Take turn on the screen rotation and here00:49:53
so place it like this00:49:55
it will be better because many people are watching00:49:57
That's how it works out a little bit00:49:59
it is inconvenient for someone to turn the phone over00:50:01
It will become possible in this way00:50:03
a little easier and cooler than before00:50:05
that after all it is all horizontal00:50:07
position, turn the phone like this00:50:09
try it, I think it should help00:50:13
Okay, once again, the beginning of the formula is the derivative00:50:16
the first part of what I do is this elevates00:50:19
denominator to the square and continue to work with00:50:22
with this thing, I first take the derivative00:50:25
I multiply the numerator by the denominator - vice versa00:50:28
I simply multiply the numerator by the derivative00:50:31
denominator and works with that00:50:34
it is visible Well, good If it is visible so from00:50:37
derivatives00:50:38
and you and I finished the product00:50:42
now we move on to the derivative00:50:46
composite function it is very important00:50:48
See what it is generally composed00:50:52
a function is a function that contains more00:50:55
one function, for example, see function00:50:59
is equal there to FX = the root of x - 900:51:04
This is a compound function. If only it were simple00:51:08
function there FX =00:51:11
√x is an ordinary function, but here in the given one00:51:16
case there is another under the root00:51:20
own special function separate function x -00:51:24
9 function is a composite external function00:51:29
root and under the root is not just x00:51:33
and there is its own separate function, and that's already -900:51:38
a completely different function, respectively, is given00:51:41
the function is complex because there is still under the root00:51:44
one of its own separate function further00:51:48
effect x = cos x² If only it were simple00:51:53
cosine x everything is fine but in arguments00:51:57
cosine has its own separate function x²,00:52:02
this is a composite function here two functions00:52:05
functions as cosine and in arguments00:52:08
cosine00:52:11
understand more or less00:52:13
I haven't finished the explanation yet. Wait and00:52:16
the cosine argument has its own separate function00:52:21
x^2 If it were just the cosine of x that is00:52:25
ordinary function but under the cosine00:52:29
a separate function is different00:52:33
x². is a composite function further EF is equal to00:52:38
X component to the power of 3x If it were simple00:52:42
exponent to the power of x, everything is fine, but you00:52:46
exponent index contains its own00:52:49
a separate function of 3x is a composite function00:52:55
figured out more or less what it is composed of00:52:58
the function is the derivative of the cosine of 2x, too00:53:02
composite function if two cosines x is00:53:07
not folded because it's a simple two00:53:09
coefficient A if the function cos 2x is00:53:14
composed because in the arguments of the cosine00:53:17
its own separate function 2x There is not just x a00:53:22
there is a separate function I'm already different 2x that's it00:53:25
not added because the two is just a coefficient00:53:29
here it is folded because 2x is separate00:53:34
the function that stands under the cosine00:53:40
Now let's talk about the derivative, what it is00:53:43
I more or less figured out how to find a derivative00:53:48
composite function00:53:50
the derivative of the composite function is equal to00:53:53
derivative product00:53:56
of the external function to the derivative00:53:59
internal function Now we are with you00:54:02
let's practice on an example, see now00:54:05
for all F from X =00:54:09
√x-9 first you need to clearly understand where00:54:13
outer function where inner function00:54:17
the outer function is what you see00:54:19
first what is located first what you are00:54:24
see, let's look first00:54:26
is equal to the square root of x - 9 times00:54:31
I look at this function, which is the first thing I see00:54:35
first I see the root so accordingly00:54:38
the first thing I do is look for the derivative of the root00:54:42
next I see the derivative next I see what's next00:54:47
I look under the root and look under00:54:51
the root is the function x - 9 the function x -00:54:56
9 is an internal function, this is what I see00:55:00
it is already in the description of the external function00:55:02
internal let's practice on00:55:06
in the second example with cosine, it is a little00:55:08
will be simpler than the first and so F from x =00:55:12
cos x² we will now take the derivative with you00:55:18
derivative of the outer function When I see00:55:22
the function cos x² that I see first I00:55:26
I see the cosine first. I'm looking for the derivative00:55:30
cosine derivative of cosine according to the table is00:55:33
- sine is written like this minus sine x².00:55:38
minus sine x². I don't care what's there00:55:42
stands for the argument the first time I see it00:55:45
cos Everything I take the derivative of the cosine derivative00:55:49
cosine - sine stood x² and still stands00:55:53
this for me is equal to multiplying and00:55:56
I am looking for the derivative of an inner function00:56:00
the first thing I saw was the cosine. Now I00:56:04
I look in. And what is there under me?00:56:07
by that cosine and under that cosine in me00:56:11
stands for x², so I'm looking for the derivative00:56:16
x squared00:56:18
the derivative of x squared is the derivative00:56:22
here is the derivative of the external function00:56:25
internal function the first thing I see is this00:56:28
cosine I find the derivative of cosine00:56:31
I look at what is under the cosine00:56:34
I find the derivative of the cosine argument00:56:39
x². = here we continue to write Well, here we go00:56:44
you and I found a derivative, so it will be00:56:47
Sin x squared is the already found derivative00:56:49
sin x² and we look for the derivative of x in the second00:56:54
powers of two in front as a factor00:56:56
it climbs out like that and I reduce the degree by00:56:59
the unit was x in the square root00:57:04
x in the first degree and we culture a little00:57:07
I multiply 2x by the whole, here is this expression - sin x²00:57:12
will be00:57:13
-2x multiply by sine x squared by00:57:19
sine x² we start with the external function00:57:23
then we look at what is in the middle00:57:26
and we look for the derivative of the inner function00:57:30
Why near the sine of x². So this is the same argument00:57:33
I had the cosine of x² and I write it as sin00:57:37
x² - everything, and how without it, this is the same argument00:57:40
maybe just all sine cosine v sine and v00:57:43
cosine must be an argument here is here00:57:45
this sin² is the derivative of the inner function00:57:49
i is the derivative of the external here is the derivative00:57:52
of the external function is multiplied by the derivative00:57:55
of the inner Here is my derivative of the inner00:57:58
functions00:58:00
the first expression we have now with you00:58:02
a very similar example from ZNO and there I am for you00:58:05
I will explain how to search. Here are the derivatives of00:58:07
that Well, it's a bit irrelevant now00:58:09
that there will be such an example anyway00:58:10
so now I will explain simply by example00:58:12
with the external examination a little more interesting like00:58:14
find the derivative of a composite function when00:58:16
root and something standing under the root00:58:20
m So let's go on, look at this00:58:23
Foreigners are very often loved very much00:58:26
give the derivative of the composite function where in00:58:30
power of the exponent something is worth it very much00:58:33
it is important to understand how to find the derivative00:58:36
exponents in not just in the X power00:58:40
that this is a normal function that is in the table00:58:43
and here the power has its own separate function -2x00:58:47
the derivative of which must be found first that i00:58:51
i do I am looking for the derivative of the external function that I00:58:55
see first first I see the exponent in00:58:58
I'm looking for a derivative00:59:01
exponents to some degree I know what00:59:04
exponent in any degree and00:59:06
will be an exponent in the same power is00:59:09
such I write the exponent to the power of -2x00:59:12
to multiply00:59:14
I look in. And what is there in my pocket?00:59:19
exponents are exponents and in the exponent00:59:22
the exponent stands for its own separate function00:59:25
-the 2nd derivative of which I need to find now00:59:29
I'm looking for the derivative of an inner function00:59:33
the derivative of the function -2x = is equal to the derivative00:59:39
-2x What is this -2 because -2 is this00:59:43
coefficient and derivative x = 1 will be -2 and -200:59:48
* E exponent to the power of minus 2x everything00:59:53
the option would be plus point00:59:57
the outer function is what you see first01:00:00
first I see the exponent in the power I know01:00:04
that the derivative of the exponent is in power so and01:00:07
will be an exponent to the next power i01:00:09
I'm looking in. And what's in it for me?01:00:13
power and in that power stands its own01:00:15
the separate function -2x is intrinsic01:00:19
function i behind, I multiply its derivative by the derivative01:00:22
external to the derivative of internal I01:00:24
I find my answer01:00:27
the brain boils a lot more01:00:30
the geometric content is still mechanical and still with01:00:33
they didn't finish with these and that class ago01:00:35
hang on hang on With the last of your strength01:00:37
let's go a little further, you01:00:42
today someone asked me about01:00:44
now we will look for the root01:00:46
Look at the derivative of the root and I'll tell you right away01:00:49
it seems that there is no derivative in the plate01:00:52
root Yes, it is not here, but it exists in01:00:56
in principle, they don't give it to the external examination either, but I do01:00:59
I will tell you now, look at the derivative01:01:01
the root of X is the derivative of the root of X is the derivative01:01:05
the root of x = 101:01:09
divide by two roots of X by 2√x Well, no01:01:15
is equal to and the derivative so the derivative is all01:01:18
the derivative of the root from x is a unit01:01:23
two roots from x in the tablet there is no Ya01:01:27
it is such a particularly derivative that once upon a time01:01:29
sometimes they don't write signs for me01:01:31
no, but you will record it for yourself01:01:33
divide the derivative and the root of x = 1 by 2√x01:01:41
now let's go to what material is there01:01:46
I don't know, as if there isn't there, there isn't. There is01:01:50
discovery01:01:52
there are already cheat sheets, cheat sheets as if there were me01:01:55
I'll look at it, it's strange, as if I didn't give it, well, okay01:01:58
well, I'll look it up later and let's go01:02:01
that the derivative of the composite function is the root of and01:02:05
under the root is its own separate function and01:02:09
so I see the first first I see the root01:02:13
that is, I take the derivative of the root derivative01:02:17
the root is recorded here, I write one share01:02:21
by two roots01:02:23
I am interested in what is under the root01:02:26
the derivative of the root is the derivative of the root of one01:02:28
divide by 2 such roots 19 -5x 1901:02:34
- 5x once again the first thing I do is look01:02:39
to the external function The function is external01:02:42
the function is the root finds the derivative of the root01:02:45
divide one into 2 such roots01:02:49
multiply now I'm looking And what's there01:02:52
stands inside under that root and under01:02:56
that root of the middle is 19 minus 5X01:03:01
correspondingly looks for the derivative of 19 minus 5X01:03:06
text is the derivative of the external function Here01:03:10
I have it as a derivative of an internal function01:03:13
Here she is, the first time I see it01:03:16
root Here I found the derivative of the root further01:03:19
I look in. And what is internal under the root01:03:23
function a under the root of 19 -5x I am looking for01:03:27
they went on his way01:03:29
divide one into two roots and leave it01:03:34
19 minus 5X is the already found derivative so01:03:38
I do not touch the external function01:03:41
we deal with the derivative of the inner01:03:44
function 19 derivative - this constant will be 001:03:50
tax 5 - this coefficient will be -5 and the derivative01:03:55
x is a unit correspondingly derived01:03:58
Here is the whole expression 0 - 5 = -501:04:02
the derivative of this expression is -5 = -501:04:06
all that was asked, find the derivative in Points01:04:10
x0 = 3, I already know how to find the derivative of c01:04:15
points instead of X, this is where I put them01:04:18
triple And I find the value of the derivative01:04:20
point is equal to we have01:04:23
what01:04:24
divide one into 2 roots with01:04:30
19- instead of X, I put a three here01:04:34
triple 1501:04:37
5*3 will be 15 1901:04:41
-15 and multiply by -5 I'm just mine01:04:46
derivative instead of X I put a three01:04:50
They asked to find the derivative at Points x0 = 301:04:54
= 101:04:57
/ 2 roots of which 19 -15 will be 4 great and01:05:04
multiply by -5 good good equals01:05:10
1 divide by the square root of01:05:14
four is 2 next I have two to multiply01:05:18
by 2 will be 4 1/4 multiplied by -5 one01:05:23
the fourth I multiply by -5 all the answers to01:05:27
ZNO in tasks with a short answer You01:05:31
you must write in the form of decimal fractions01:05:33
everything that happened there, who is messing around there01:05:36
everything is fine everything is fine And so 1/4 I01:05:40
multiply by -5 and write in the form01:05:44
the resulting decimal fraction will be minus01:05:51
1.25 It seems so so one whole 25 hundredths01:05:56
we answer everything with this number and01:06:00
we earn a plus point on the external examination, well, what can I say01:06:03
you don't have to find something like supernat01:06:06
derivative to understand that it is not easy01:06:09
function01:06:11
made to replace X with a triple and01:06:15
count a specific number01:06:19
tax why 1.25 if I multiply the fraction by the whole01:06:24
number is this whole number I multiply by01:06:27
the numerator of the fraction, the numerator is 101:06:30
multiply by -5 will be -5/4 Now as -5/401:06:36
convert to a decimal stupidly in the Y column01:06:39
I simply divide column 5 by 4 and find01:06:43
what it is01:06:44
1.25 I take one at a time and subtract four01:06:49
I put the remainder 1, add 0 and divide01:06:53
until the quotient is divided by the remainder by zero01:06:55
will be 1.2501:06:59
there will be no infinite fraction on the final exam01:07:02
select numbers in such a way that01:07:04
there will be no infinite fraction Everything01:07:07
Every year at the ZNO there is a short publication01:07:09
the answer will be converted to decimal without01:07:11
infinite of these particles with ease01:07:13
will be translated and you write all the answers01:07:16
they are unusual only by the decimal fraction01:07:19
God help me to write in the boxes01:07:21
3/4 somehow That's how it is because my children are like that01:07:24
chudyli no we convert to decimal and01:07:28
we record01:07:29
Well, we've figured it out. Now let's move on01:07:32
we have two more aspects of it now01:07:36
the geometric content of the derivative is mechanical01:07:40
the content of the derivative after that click And I01:07:44
I let you go, let's start with the geometric one01:07:47
geometric content of the derivative01:07:50
someone there will already write early01:07:54
they didn't get the topic to the end And so01:07:56
let's start with the geometric content01:07:58
derivative, see I have a function of some kind01:08:02
here it is drawn in red01:08:04
now I am highlighting it here is some function y =01:08:08
FX I select point point x0 and in this01:08:14
I draw the point to the function tangentially like this01:08:18
the value of the derivative at this point and F301:08:24
from x001:08:26
=01:08:28
the tangent of the angle of the tangent to the axis01:08:32
tangent here the tangent formed the angle Alpha01:08:36
so here is the tangent. This is the corner of Alpha01:08:41
the value of the derivative function01:08:45
at point x0 And point x001:08:50
this is the Point at which a tangent was made01:08:54
since there is a function y = FX Well, there is something there01:08:58
any choose a specific point x0 This01:09:03
just a designation of a specific one01:09:05
point and at this point the functions are carried out01:09:10
tangent This is the tangent of the slope01:09:15
the angle that forms a tangent to the x-axis01:09:19
here is the angle alpha is equal to01:09:22
to the value of the derived point x0, i.e. in that01:09:28
the point at which this tangent was carried out What not01:09:32
of course01:09:35
I honestly don't know what to do01:09:38
it is not clear, everything is fine, let's give it one more time01:09:41
Once I explain, the function y = FX is selected01:09:46
some point on this function, well, absolutely01:09:48
any, here, this, and marked x0 x0 this01:09:54
just some specific point there 3-4-5 and01:09:57
so on, the tax was carried out related to the function01:10:01
at this point and through this point is selected01:10:04
spend tangentially so here is the corner01:10:09
forms a corner tangent to the x-axis01:10:12
Alpha Tangent Here is the alpha of this angle01:10:16
is equal to the value of the derivative point x001:10:20
that is, at the point where this tangent was drawn01:10:23
everything you need to know about geometric01:10:26
the meaning is that the derivative of the point is the value01:10:30
tangent01:10:33
we look at the angle of inclination of the tangent, for example01:10:36
And it's still early, for example now with this one01:10:41
the geometric meaning of the equation follows01:10:45
tangent if I have a function and I know01:10:49
the point at which this tangent A was drawn01:10:53
tangentially it is some straight line I can01:10:56
write down the equation of this straight line01:11:00
i.e. the tangent that was drawn to mine01:11:04
functions at the points min of the equation of the tangent01:11:08
it seems to be in the reference materials too01:11:11
you and I will train today how01:11:14
write here the equation of the tangents And what with01:11:16
him to do Please don't start01:11:18
It's not difficult to panic. Maybe in theory01:11:21
now it looks scary but on01:11:23
practice is very easy for me01:11:25
history was with higher mathematics i01:11:27
I went to lectures and listened to this theory and no01:11:31
the devil did not understand but came to01:11:33
practice and solve completely Well01:11:35
everything was fine with me, that is01:11:37
the theory looks scary but don't panic01:11:40
please, don't shout that nothing is wrong01:11:41
understandable I will become a suicide, they collected what01:11:44
suicide, guys, well, these are easy things, that is, on01:11:47
in theory now it's scary, but in practice01:11:50
now it will be easy for you Believe now01:11:52
let's look at the examples, I think you will understand01:11:55
the equation of the tangent looks like this01:11:58
so y = function so y = Comfy expression x01:12:03
F dash from x0 that is this value01:12:07
derivative at point x0 you already know01:12:10
you can also calculate the derivative in points01:12:13
It was easy. That is, you count the F stroke01:12:16
from x zero multiply by X - x001:12:21
x0 - This is this particular point Yaki01:12:24
draw a tangent plus F from X zero01:12:28
that this is the value of the function at the point x0 you01:12:33
simply substitute a specific one into the function01:12:36
point x0 is a specific number and you count it01:12:39
this is F from X of zero, let's say for example01:12:42
it will be a little easier to see an example with01:12:47
ZNO01:12:48
In 2007, there is a tax function and a function01:12:54
draw two tangent points X1 here is the point01:12:58
X1 and at the point X2, here is the point X201:13:03
find the value of the expression01:13:06
F stroke from X101:13:11
from X2 what we know is the geometric meaning01:13:16
derivative and we clearly understand that01:13:19
the value of the derivative point is the tangent01:13:22
the slope of the tangent angle is all you have01:13:25
to understand that the value of the derivative point is This01:13:28
just the tangent of the angle of inclination of the tangent to the axis01:13:32
point X1 Here it is F stroke from X1 is tg01:13:39
angle of inclination of the tangent, let's look for this one01:13:43
the tangent here is an angle of 45, respectively, here too01:13:47
angle 45 because these two angles are vertical01:13:52
equal to each other, that's what this cat is01:13:55
forms a tangent to the x-axis01:14:00
we drew the points x0 x1 so here is the corner, which corner01:14:06
45 degrees, respectively, the value of the derivative01:14:11
at point X1 it is a tangent of 45 degrees, that's all01:14:16
let's look01:14:18
F stroke from X1 to X1 is equal to the tangent01:14:26
45 degrees because the tangent that was spent01:14:30
at the point X1 forms an angle of 45, that's all01:14:35
the value of the derivative point X1 is tg 4501:14:40
degrees according to the tangent plate 45 That's what01:14:45
such This is one01:14:48
further01:14:51
F stroke from X2 is the tangent of the angle of inclination01:14:57
tangent drawn to point X201:15:01
the tangent of the angle that forms a tangent with01:15:05
the x-axis is a question for you, what is the angle01:15:08
forms Here this is tangent to the x-axis Ta01:15:12
it does not form any angle other than 90 with the axis01:15:15
igrikov 90 and with the x-axis it is01:15:19
there is no angle parallel to it, of course01:15:22
I'm not an igrikov from the 90 igrikovs, but I am01:15:27
I ask about the x-axis, it is parallel and01:15:30
the truth has no angle, that is, Some01:15:33
and there is a tangent to calculate Well, there is nothing to do either01:15:36
respectively, the value of the derivative function01:15:40
function at the point X2 = 0, there is no angle there01:15:45
there is no tangent because F3 F is a stroke from x01:15:50
the second from X2 = 0, well, there is none there01:15:55
there is no angle there, the tangent is zero, it is F3 from01:15:59
X1 is a unit, F a stroke from X2 is a zero01:16:05
one plus zero will be 1 option A + in you01:16:08
they did everything, why exactly the tangent is like that01:16:13
the geometric meaning of the derivative is the tangent01:16:16
angle of inclination of the tangent01:16:19
why, once again, because this is a tangent01:16:22
which was carried out at the point X2 does not form01:16:26
there is no angle with the x-axis there01:16:29
angle there is nothing to calculate the tangent from01:16:32
this tangent visually looks parallel01:16:35
see x if Two lines are parallel01:16:38
Well, what is the angle there, there is no angle between them01:16:40
no truth they do not cross between01:16:42
itself, respectively, that tangent Well01:16:45
there is nothing to calculate the value of the derivative01:16:48
point X2 = 0 there is no angle, just z01:16:52
point X1 everything is fine here it forms a tangent01:16:56
the angle is 45 and according to the geometric content01:16:59
value of derivative point X1 = 45 tangent01:17:04
45 and tg 45 is a unit, everything is cool01:17:08
counted01:17:10
exhale, let's go on, that's it01:17:14
cool task You will have it in01:17:16
average level And when I gave it01:17:18
I had students last year01:17:20
there are many questions about how to do it01:17:23
such a task, see is asked which one01:17:26
forms an angle with the positive direction of the axis01:17:30
x is tangent to the graph of the function ax01:17:33
such at the point x0 = -1, that is, it is necessary01:17:38
find a specific angle in degrees which01:17:41
forms a tangent drawn to Here is this01:17:44
function graph01:17:46
at the point x0 = -1 somehow draw there01:17:51
count with protractors. What is the angle there?01:17:54
it won't work, what should we do, I can01:17:58
find the derivative of the function so substitute x001:18:03
and find the derivative of the function at the point x0 Я01:18:08
You have already done it many times today01:18:10
even I was told that it is easy that F01:18:12
the stroke from x0 tax can be easily calculated01:18:16
If I find the derivative now, I will substitute x0 I01:18:21
I will find the derivative of the point x0 that I will find I01:18:26
I will find the tangent in the geometrical way01:18:28
content I will find tg here is this Alpha I01:18:32
find tg of the angle formed by the tangent01:18:37
the positive direction of the x-axis, and I can01:18:40
I can tell by the tangent what angle it is01:18:43
degrees can When I see tange Am I01:18:47
I'll tell you what corner it is, I'll tell you, let's go and look for it01:18:50
derivative y dash = derivative of such a function01:18:55
one fourth is our coefficient01:18:58
we touch and x in the fourth we count 401:19:02
climbs ahead as a multiplier ahead as01:19:04
the factor and the power of x decreases by01:19:07
unit was x^401:19:10
will become x to the third power is equal01:19:14
1/4 is shortened, x remains in the third01:19:19
power all the derivative found a class on01:19:22
derivative By the way, x^3 turned out very well01:19:25
substitute in the function instead of X - 1 for01:19:31
in order to find the value of the derivative in01:19:34
at point x0, the state tax will be instead of X - 101:19:39
-1 to the third power minus one in01:19:43
to the third power is -101:19:46
that is, tg = -1 the tangent of the angle it forms01:19:53
the text is tangent to the positive direction of the axis01:19:57
x is equal to -1 and now the question is for you01:20:01
what angle does tg -1 give me01:20:06
clever first if tg is negative - this01:20:11
only an obtuse angle only an obtuse angle gives01:20:15
I have a negative trigonometric function of 3001:20:18
45 does not fit, always have sharp corners01:20:21
there are positive sines, cosines, tangents and01:20:25
cotangents01:20:26
negative have only obtuse angles And now01:20:30
let's go back to trigonometry01:20:34
calm down, you are not stupid, you are all smart01:20:36
to trigonometry and remember the formulas01:20:39
summary I am writing you a tangent tangent01:20:44
18001:20:45
180 - Alpha is also the same as minus tangent01:20:53
alpha minus tangent Alpha I have it now01:20:57
minus tangent alpha equals -101:21:12
[music]01:22:02
it forms a tangent of 135 degrees01:22:08
exhale01:22:10
where does 45 come from I have minus one Yes - I don't01:22:15
I pay attention because these are summation formulas there01:22:17
and there should be a minus, I'm just talking about01:22:19
unit, the tangent of which angle is unit01:22:23
45°. so alpha alpha equals 45,18001:22:29
- 45 will be 135 is By the way expansion01:22:33
signs and there immediately write Tam01:22:35
tangents of obtuse angles and there are such plates01:22:38
Where they write that the tangent of 135 is -1 but on01:22:43
ZNO does not give such a sign, they give it there01:22:45
this is more concise and there are blunt tangents01:22:48
corners are not given01:22:50
exhale a little more now we are with01:22:54
you are practicing how to write equations01:22:58
the tangent before that is a little light01:23:01
a task from the external examination and in the next one already01:23:03
we will practice how to write equations01:23:06
we went on a tangent, we talk about a tangent and01:23:10
about Laitov's tangent equation Yes, that's it01:23:13
Laitova calmed down so much, everything is okay01:23:16
everything is fine01:23:18
we read, indicate the equation of the line that can01:23:23
be tangent to the graph of the function there01:23:26
of course And so y = FX at the point x0 = 201:23:32
if f'01:23:35
= -301:23:38
Now we are looking carefully at01:23:41
I know the equation of the tangent here01:23:44
under the condition that F is a stroke from x of zero01:23:48
is equal to consider exactly -39 taxa -3 F01:23:53
stroke from two. And my two is x001:23:56
so that is, F is a stroke from x0 = -3 by condition01:24:03
we return here and write -3 and now01:24:07
let's think if I'm right now x0201:24:12
well, but if I am now01:24:15
I open the brackets that I have my first01:24:20
factor I have F stroke from x0 from x01:24:27
of zero is multiplied by X when i01:24:30
I open the brackets, I clearly see the meaning01:24:33
of the derivative point x0 will stand near x01:24:38
multiplier respectively I am now looking for that one01:24:41
option where there is -3 next to x next to x01:24:46
the value of the derivative point x0 should be in01:24:52
me under the condition of the value of the derivative in points01:24:55
x0 is -3, so I'm looking for that option where01:25:00
next to x is the coefficient -3 And here it is01:25:04
only one only in one version01:25:07
next to x is the coefficient -3 next to x01:25:12
the tangent formula must contain a function01:25:16
the derivative of the function of the point x001:25:19
accordingly, they are looking for that option where near01:25:22
x is -3 because minus 3 is the value01:25:26
derivative at the point x001:25:31
Again and again when I look at the equation01:25:35
I'm off to a tangent and I'm starting to open the brackets01:25:39
F is the stroke from x of the zero product of X01:25:42
accordingly, you are in my tic equation01:25:46
next to x should be F3 from x0 ago01:25:51
among all options I am looking for that option01:25:54
where there is a -3 next to x, because F is a stroke01:25:58
from x0 it is -3 by the condition near x has01:26:05
the value of the derivative point x0 is in me01:26:09
under the condition of the value of the derivative at the point x001:26:13
it's -3, so I'm looking for an option where near x01:26:17
there is a number -3, this is the d + point option01:26:20
did01:26:22
Let's exhale. Yes, you can not solve it01:26:25
you will not immediately write clearly here01:26:29
equation of the tangent because here01:26:31
not enough data to write01:26:33
There you can completely equation the tangent01:26:36
find it Only Thus here and01:26:39
the question may be because it is exactly here for you01:26:43
not enough data to write the equation clearly01:26:46
tangent only in this way01:26:49
the buttons will be about the geometric content01:26:52
but it will not be about a tangent like this01:26:56
we are literally 4 more now01:27:00
slideshow including this click01:27:03
we disassemble the clicker, I let it go somewhere01:27:06
I'll let you go in about 15 minutes01:27:08
a little bit Well 10 even I think so Now01:27:10
you and I will clearly make up already01:27:13
the equation of the tangent I went to the function y =01:27:18
x³ here the assignment is not to be taken from ZNO And here01:27:23
I really don't like the options01:27:26
there are so many answer options here01:27:28
Stupidly written, you can't think of anything stupider01:27:31
therefore, in principle, no options01:27:33
get oriented We will do it a little differently01:27:36
there is a function y = x³ - This is the most difficult next01:27:41
the mechanical content will be simpler01:27:43
it's much simpler there, in general, everything is there01:27:45
very easy there three slides click01:27:48
lets you suffer a little bit01:27:50
suffer a little at all is a function of y01:27:53
= x³ you need to write the tangent01:27:56
of this function there is a point like this with01:28:00
coordinates 2 and 8 when you see01:28:02
the coordinates of the point are the first digit01:28:06
corresponds to x, respectively, x0 in the data01:28:10
case is equal to 2 I have to write01:28:13
the equation of the tangent point x is zero which01:28:18
is equal to 2 all the first digits in the point01:28:21
is responsible for x means x is zero01:28:25
is equal to 2 all I am here now for that01:28:29
in order not to turn over there, I write the formula for01:28:32
the equation of the tangent Y =01:28:36
F stroke from x zero I write you down01:28:39
the formula for the equation of the tangent to x01:28:42
of zero is multiplied by x - x0 x - x001:28:48
dachshund and plus01:28:53
fx0 Now we will gradually be with you01:28:57
look for all these pieces that are in01:28:59
substitute formulas and everything is easy and cool01:29:02
it turns out, you can start by finding the axis01:29:05
this piece of F x zero is stupidly U01:29:09
instead of X, I put a two because my function01:29:13
x0 = 2, that is, I put a two and I find01:29:17
a function from x of zero It will be what it will be01:29:21
2³ = 8 all F from x0 = 8 further F stroke from01:29:29
the value of the x of zero must be found01:29:31
of the derivative at the point x0, what should be done?01:29:34
it is obvious that to begin with it must be found01:29:38
derivative so that in the cube Well, by convention01:29:42
of functions y = x³, we all set x³01:29:47
. why is x0 again at a point for the first time a number01:29:51
is responsible for x 2 for player 8 no01:29:54
refers to x 8 refers to the player01:29:57
the first digit corresponds to x means x0 =01:30:01
2 now we are looking for the derivative of the game stroke01:30:05
is equal to Well here the derivative is so01:30:07
x³ 3 in advance as a factor of the value of the power01:30:11
let's reduce by one it was in the third01:30:15
power will become in the second power it01:30:17
the derivative of I is now in the derivative instead of X i01:30:21
I put x0, I put two And I find01:30:24
is equal to will be 3 *01:30:29
2 squared two squared instead of X01:30:33
put the number 2 is equal to 3 will be multiplied01:30:38
by 2² there will be 4 3 * 4 = 12 the whole value01:30:44
I found the derivative point x001:30:49
12x will be X x zero equals 201:30:53
I have all the values I substitute01:30:56
I open the parentheses and simplify a little there01:30:59
transform and have a beautiful beautiful one01:31:01
the equation of the tangent went to = and I write01:31:06
the value of the derivative function at the point x001:31:11
found it 1201:31:14
multiply by X - x0 x0 is two x01:31:20
minus two plus values are just functions01:31:23
points x0 is eight plus eight let's01:31:27
open the brackets and transform a little more01:31:30
to make it even better, we will multiply 12 by X01:31:33
12x 12x 12 multiplied by -2 will be -24 -24 and01:31:43
+ 8 + 8 =01:31:47
we have 12x 12x - 24 -8 will be -16 so01:31:56
so that's minus 16, that's all. Here's mine01:32:00
the equation of the tangent y = 12x - 16 here01:32:06
the options are written very stupidly. I don't even01:32:09
want now Thus convert to01:32:11
reduce to one of the options because they are here01:32:12
Well, straight to the maximum, well, strangely recorded01:32:14
the equation of the tangent end y = click01:32:19
will be today 12h - 16h 12h - 16h if01:32:24
to figure it out, this is an option, yes, I agree01:32:27
but I don't want to convert because it01:32:29
Well, in a completely stupid way, it is written there like this01:32:32
smart people write y = 12x - 1601:32:37
all01:32:39
once again you are looking for the derivative look for it01:32:42
find the value of the function at the point x001:32:46
the value of the derived point x0 in the formula all01:32:49
substituted opened parentheses converted01:32:52
simplified everything, everything turned out great01:32:54
cannot be reduced by 2 because it01:32:57
a function is not an expression, I can't01:32:59
now take and divide both particles01:33:01
by 2 right here y stands for shortening by two01:33:04
here will not translate the Lord this function here01:33:07
nothing is reduced and remains so01:33:09
from where three they ask from where three I explain01:33:13
bunny look at x³ 3 ahead as a factor01:33:16
when looking for the derivative and the sign of the exponent01:33:18
decreases by one was in the third01:33:20
to the second degree, everything became second degree01:33:23
now exhale the mechanical meaning two01:33:29
examples click and release More01:33:32
just a little bit and I have a By the way function01:33:34
I prefer to take a taxi01:33:35
to go home because it's not enough for three01:33:38
I'll have time to sort out our derivative01:33:41
we drove to the end, see about the reason01:33:44
of mechanical content yes cool question to01:33:47
text two put it physics actually01:33:49
Yes, it's true. It's physics, but mechanical01:33:53
the content of the derivative is really related to physics01:33:55
But in order to work with him, you have to01:33:58
to look for a derivative precisely because of the mechanical one01:34:01
the content of the derivative Although it is a physical particle01:34:04
what is there even though there is a physical particle01:34:07
we need to study it and deal with it01:34:12
due to the fact that they ask about it at the final exam01:34:15
mathematics See you should not arise01:34:18
I will try to find out what it is there01:34:20
material point, what movement is there01:34:22
rectilinear or curvilinear which us01:34:24
the difference, all you need to know, see U01:34:27
you have an equation that describes the motion01:34:30
of some material point or another01:34:32
there body, it will be given to you in the condition01:34:34
problem If you take the first derivative01:34:38
from this equation you find the instantaneous01:34:42
speed what is instantaneous speed it is01:34:45
specific speed less time01:34:47
for example, I'm calling, but I'm Lizochka writes about01:34:49
asks About the police. I call the bells and whistles01:34:52
I say Lizochka, where did she go to school yesterday01:34:54
tell me Ot U 8:20 which one you had01:34:59
the speed of the fleece, says Christina, is 3 km/h01:35:02
That is, this is the speed at a specific moment01:35:05
of time at a specific time, it was like this01:35:07
velocity is the instantaneous first derivative of01:35:11
the equation of motion describes to you the instantaneous01:35:14
the speed of the second derivative describes the instantaneous01:35:18
acceleration I will not succeed now01:35:20
explain in detail what acceleration is01:35:22
and so on And so on the first derivative is01:35:25
the instantaneous speed of the second derivative is the goal01:35:29
acceleration how to find the second derivative then01:35:31
we take the derivative twice, we took the single derivative01:35:34
derivative and the second derivative again everything01:35:36
the first derivative is the instantaneous speed01:35:39
the second instantly accelerates immediately like this01:35:41
insight into external examinations is very rare when asked about01:35:45
instant acceleration of I for the entire history of the ZNO01:35:48
saw about instant acceleration once01:35:50
the question about instantaneous speed is asked01:35:53
are often asked regularly why now with her01:35:55
we understand two tasks click and I01:35:58
I'm letting you go, let's go for a task from the external examination01:36:01
about instantaneous acceleration, see what I01:36:05
told you You have a given equation of motion01:36:08
here is the equation given by the condition01:36:11
It is necessary to find And by the way acceleration so01:36:15
see need to find the acceleration in01:36:17
the moment of time t is equal to 10 acceleration -01:36:21
this is the second derivative. I take two times the derivative01:36:24
replaces the trail with 10 I01:36:26
I find the acceleration at time T01:36:29
is equal to 10, we take the derivative twice01:36:31
put 10 and everything turns out great, let's go01:36:33
we are looking for the first derivative S stroke from t01:36:37
now with you the first derivative is equal to 2/301:36:41
We don't touch the coefficient, we look for the text01:36:44
derivative t in the third power 3 forward01:36:47
as a multiplier with a smaller exponent on01:36:50
unit was t in the third power becomes01:36:53
t in the second degree then -2 *01:36:57
We reduce t² 2 in advance as a multiplier01:37:01
the indicator per unit was that in the second01:37:04
power becomes simply t and the derivative of four01:37:07
T will be simple + 4, we transform it a little01:37:10
to look a little better01:37:13
3-3 will be reduced to 201:37:16
2t². 2 t² - 2 * 2 will be 401:37:22
-4t and + 4 This is my first derivative01:37:26
instantaneous acceleration is the second derivative01:37:29
accordingly I now Once more Here01:37:33
I take the derivative of this expression and look for the second01:37:36
I am looking for the derivative, the second derivative, the second time01:37:39
I take the derivative already from my formed one01:37:42
expressions went here with Here is this expression01:37:45
the second time I take the derivative will be 2 * t²01:37:50
we reduce the two in advance as a multiplier01:37:52
the indicator per unit was that in the second01:37:56
power becomes that in the first or simply that01:37:59
the derivative of -4 T will simply be -4 and the derivative01:38:03
plus four will be 0 because it's a constant01:38:06
is equal to 2 * 2 So why01:38:09
not in zero01:38:12
4 is a coefficient and the derivative is also equal01:38:16
units 4 to 1 will be 4 all01:38:20
the derivative of the constant is zero, forget what01:38:23
yes yes derivative constant is zero I plus01:38:26
zero is no longer written derivative minus 4 t -4 -01:38:30
this derivative coefficient is also a zero derivative01:38:33
something is a unit all -4 to 1 will be 401:38:36
went two by two four T - 4 it01:38:41
an equation that describes me instantly01:38:43
acceleration is asked about instantly01:38:45
acceleration at the moment of time t = 10 i01:38:49
instead of the path, I put 10 and find it01:38:52
acceleration at a time of 10 seconds for what01:38:56
twice instantaneous acceleration a instantaneous01:39:00
acceleration is the second derivative is two01:39:02
times We take the derivative in order to01:39:05
get the equation by which it is described01:39:07
instant acceleration01:39:09
creation is the second derivative went01:39:12
I bet four will be multiplied by 10 and -401:39:16
4 * 10 40 minus 4; 6 option in plus shaft01:39:21
made it easy or have any questions01:39:25
why minus 4 and not plus 401:39:28
bunny look it's about This is it01:39:30
here is this piece minus 4 and + 401:39:34
the derivative is the Nolik constant plus 0, i.e01:39:39
I wrote it because it makes some sense01:39:41
the derivative of the constant is zero, this is -401:39:44
What Here is a derivative of Here01:39:46
expression minus four T and the derivative - 4 T01:39:50
it will be -4 one second my mother will call me01:39:53
or worried about where I disappeared, I will answer and01:39:55
I will say that we are okay with this01:39:58
figured out if I have any questions about the lesson01:40:00
mother01:40:04
ok, figured it out01:40:11
figure it out ok then the last one01:40:14
a click and let go a little more completely01:40:17
Don't worry, no one drops out of school01:40:20
my mother has no unquestionable island01:40:23
Very briefly Mother said, she is not a lesson01:40:24
took disconnected Well, we're fine01:40:27
the last example for today click and01:40:31
I'm letting you go, there's still a little bit more01:40:34
after all, the law of motion. Here it must be found01:40:38
value of T value of time moment of time at01:40:42
which has an instantaneous speed of 7601:40:45
instantaneous velocity first derivative I am looking for01:40:48
the first derivative F stroke from T is equal to and01:40:52
we are looking for the first derivative equal to 201:40:55
the coefficient I do not touch it where in the square01:40:58
the deuce climbs ahead as the multiplier will be01:41:02
2 multiply by and decrease the exponent sign01:41:06
per unit was that in the second power01:41:09
It becomes simply how it is in the first degree01:41:12
yes Plus is looking for the derivative three three T three it01:41:17
it's just that the derivative coefficient is also equal01:41:20
we will cultivate the units 2 by 2, there will be 401:41:24
4t + 3 4t + 3 I ask the moment of time at01:41:31
whose instantaneous speed is equal to 76 all01:41:35
So here is my equation that describes01:41:38
instantaneous speed should be equated to01:41:43
76 and find the value of the heavy moment of time01:41:48
so time as at which at which moment01:41:51
the instantaneous speed was 7601:41:54
we solve the equation01:41:57
4t = 3 transfers of the exercise, change the side01:42:01
the opposite sign will be -3 = 76 - 3 =01:42:06
73 and in order to find a little girl01:42:11
I divide both the left and right sides by 4 and have01:42:15
that it is equal to 73, we divide it by 4 and translate it01:42:20
there will definitely be a decimal fraction01:42:24
18.25 I already anticipate the question A how01:42:28
translate into column 73 I just divide by01:42:32
I take 4 one at a time, subtract 4, the remainder is 301:42:38
I lower three, take eight, subtract01:42:42
32 I have a remainder of one I put to whom I add 001:42:48
and drove the quince until the rest was gone01:42:52
everything will become Nolyk01:42:55
18.25 + points earned01:43:01
it cannot be the time of what cannot be01:43:03
time can01:43:04
everything can be, that's why time can't be01:43:08
fractional number moment of time can do01:43:11
fractional you already have time there01:43:13
8-208-30 here in seconds, that is, 18 there01:43:16
seconds and there are still 25 20 025 seconds Well01:43:21
can be a time with such a fractional number01:43:24
they Last year there was a test01:43:26
a homework problem where they also counted01:43:29
there they counted the time and you were something like 25 there01:43:33
whole 5/10 and I had questions from01:43:36
students and How can this be? Well, 25 hours01:43:38
there is half an hour, that is, 30 minutes01:43:41
i.e. it's normal time can give01:43:43
number today01:43:45
I turn everything on for you click on click01:43:48
today we have 4 questions and 4 minutes01:43:52
we deal with every question and I you01:43:55
I finally release the click button 4 question 401:43:59
minutes I turn everything on because you will tell me whether01:44:02
you can see everything like this for a second01:44:04
the derivative can be seen in the classic Oh, and why three01:44:09
is standing01:44:11
Damn it, I'll finish it in a second01:44:14
I'll change it a little because something costs me01:44:16
the time limit is 3 minutes on should be 401:44:19
Now I will put 4 minutes yes Yes and you01:44:23
he turned it on. That's all four, that's all01:44:27
four questions four minutes Everyone left01:44:32
I'm silent, you perform, then we'll figure it out01:44:36
with each question and finally let go01:44:52
it's already the end Everything is already a click01:44:55
we'll figure it out, I'll let you take a taxi for yourself01:44:59
order yet01:45:01
because something Today you and I are so Yes01:45:03
a little long01:45:50
Romochka, you are telling people the wrong way01:45:54
Don't listen to rum, it's wrong01:45:57
by the way, the answer is Roma, if you can tell me01:45:59
you at least give people the right advice01:46:09
Divide the derivative of llc unit by x01:47:24
if you have a minute, we will01:47:27
complete01:47:49
here Lera is very active asking about01:47:51
I'll do some Lerochka New Year01:47:54
like everyone01:48:00
so 20 seconds and we finish01:48:11
I think that you are writing numbers, we are counting01:48:14
pancakes01:48:18
yes, 5 seconds and that's it01:48:23
2 1 0 all I'm showing you yours now01:48:26
results Tell us about your results01:48:29
someone has 10001:48:40
thank god everything is fine now everyone01:48:44
ok then i'll end here01:48:47
the average score certainly shows that it is difficult for you01:48:49
it was today. Okay, I'm finishing and where will it be01:48:54
presentations and so Return what to return I01:48:57
I'm already nervous about the jug. I won't return it01:49:00
Okay, I didn't count my friend now01:49:03
let's figure it out and so on the First one is necessary01:49:05
find the derivative of Romochka's function there is something01:49:07
tipped you off but didn't tip you off01:49:09
this is wrong01:49:10
here difference here sum in groups ratio01:49:14
of this function that here it is important to see what01:49:16
is the derivative of the composite function of the five01:49:19
It's just a coefficient, I don't touch it01:49:21
I am looking for the derivative of the composite function sin 7x01:49:26
let's figure out the external function is sin01:49:29
what I see first is the derivative of the sine01:49:32
is equal to cosine cosine 7x I look at A01:49:36
which stands under the sine and under the sine01:49:39
standing internal function 7x function 7x -01:49:43
the inner derivative of the inner function01:49:45
derivative 7x = 7 multiplied by 7 - 7*01:49:50
x² derivative of x squared of two01:49:54
forward as a factor of the power of x with a smaller na01:49:57
unit was x² will be simply x and na01:50:01
the constant is zero, it makes no sense01:50:03
write 5 at 7:35 am 3501:50:08
cosines 7x cosines 7x - 7 * 2 will be01:50:16
- 14x all, I answer which option01:50:20
option in and I earn a point What is not clear01:50:26
Have you figured it out here, are there any questions?01:50:29
there is no right answer01:50:32
correct option in the correct answer01:50:34
everything is fine. Why is it a derivative?01:50:37
composite function is the derivative of the external p01:50:40
cosine of 7x to the inner derivative01:50:44
the inner function is the 7x derivative of 7x = 701:50:48
That's why we multiply option 7 by the second01:50:52
find the derivative of the function here too01:50:55
composite function y dash = derivative01:50:59
the logarithm of the natural derivative of LN01:51:01
is equal to what 1 divide by and the argument01:51:05
2x now I'm looking and what's under there01:51:09
my logarithm, what is the argument There is not01:51:12
just x there 2x - it's also folded01:51:16
function I take the derivative of the inner01:51:18
functions take the derivative of 2x will be simply 201:51:21
then +2 * derivative xa³ triple comes out01:51:28
forward as a factor and want an exponent01:51:31
per unit was x^3 becomes x squared and -301:51:37
- it's just if the constant has no meaning01:51:40
to count perfectly 2-2 are shortened01:51:43
it remains only to divide by x + i01:51:47
here we multiply 2 * 3 will be 6 6x² g plus01:51:53
Ball did everything here where I caught you on01:51:56
because here too the function and was composed01:52:00
here was a compiled function here is important with01:52:03
do not get confused by this and do not forget it01:52:08
all01:52:09
the third question has to be found01:52:12
yes, the value of the derivative point x0 is now i01:52:17
I will enlarge it a little for you on the phone01:52:19
then it seems that it does not increase in anyone01:52:20
on the laptop and you would now have a slide01:52:22
let's look a little closer01:52:25
drawing of the value of the derivative point x0 axis01:52:28
this is the point x0 so the value of the derivative is in this01:52:31
point is the tangent, the slope of the angle of the tangent to the axis01:52:34
this code which forms a tangent to the positive01:52:40
I just need the direction of the x-axis01:52:42
find the tangent Here of this angle and01:52:44
accordingly, I will find the value of the derivative in01:52:46
point x0 everything, how to find tg1 angle, what is it01:52:51
tangent in general trigonometry is01:52:53
relation01:52:55
it's a relationship01:52:56
of the opposite leg to the adjacent one01:53:00
cells tangent Here is this little blue one01:53:04
of a small angle, what is equal to po01:53:07
cells opposite side C1 a01:53:10
the adjacent side is two cells01:53:13
Here is the tangent of this small angle01:53:15
so it's 1/2 tg of a small one01:53:20
blue corner is 1/2 and tg from red01:53:24
is the tangent of the angle it forms01:53:28
tangent to the positive direction of the x-axis01:53:31
I need a red corner01:53:33
the red corner lies next door01:53:36
red cat is 180 degrees 18001:53:41
degrees minus the Red corner minus Y01:53:45
I write well. Thus, here is the minus of Red01:53:48
corner yes01:53:49
now according to the formulas of the reduction I have01:53:53
the tangent of the red angle opposite God and01:53:57
above the blue corner of the blue corner minus01:53:59
so everything is back - the blue corner minus here01:54:03
this little blue corner is now over01:54:06
in the summation formulas I have a tangent tangent01:54:11
180 degrees01:54:13
180 degrees minus Alpha is also the same thing01:54:20
minus the alpha tangent, respectively, in me01:54:25
the tangent of the red angle is the same as01:54:29
minus the tangent of blue. That is, it is minus 1/201:54:33
and - 1/2 is -0.5 option d You have01:54:40
to understand what the tangent equals So there01:54:43
you have to freeze because it's there01:54:44
right triangle Why did you take01:54:46
larger angle see01:54:49
in the content of the derivative, we have the tangent of the angle, which01:54:54
forms a tangent with a positive direction01:54:57
the x-axis is the positive direction01:55:00
right corner so I take the right corner01:55:03
positive numbers on the right side so right01:55:06
sides run more positive numbers01:55:09
direction 8x7 from the right side of volume i01:55:13
I take exactly that red right corner01:55:21
Okay, nothing is clear01:55:24
why on stage once again we have a geometric one01:55:27
the contents take the angle formed by the tangent to01:55:31
in the positive direction, the x-axis and are positive01:55:34
the direction from the right side all accordingly i01:55:37
I take the right corner, here is a larger number01:55:40
the positive direction accordingly and I take it01:55:43
this is the red corner01:55:45
valence if you want to ask later01:55:48
later then No problems and all the last01:55:50
question and I'm letting you go why 1/201:55:53
Because a tangent is a ratio01:55:54
the opposite leg to the adjacent one01:55:56
opposite in cells one cell01:55:58
adjacent cells are two cells01:56:00
therefore, each other is the opposite of each other01:56:03
a cell will adjoin two cells to one01:56:06
the second last question and let me go. Well here01:56:09
I think it's easy to see and after what time01:56:12
from the beginning of the movement point stopped by01:56:14
logic If the point stops then what01:56:19
occurs then there is a velocity equal to01:56:21
zero, respectively, I am looking for an equation according to which01:56:26
speed is recorded for this I take01:56:29
the first derivative is equal to the first derivative01:56:32
look for the input 2.5 and we look for the derivative of tem v01:56:37
squared derivative and squared is equal to01:56:39
two in front as a factor and a sign and01:56:42
the indicator decreased by one was the subject of01:56:45
squares are simply minus 15 t derivative01:56:49
will simply be 15 2.5 I multiply by 2 will be 501:56:55
we have 5D -1501:56:59
5t -15 is the equation by which it is written01:57:02
instantaneous speed if the point is stopped01:57:06
then the speed must be zero for all I01:57:09
I'm looking for a point in time01:57:11
stopped 5 t -15 = 001:57:15
that means 5t = 15, I - 15 I fail the exercise01:57:23
side and change the sign to the opposite01:57:25
was -15 becomes plus 15 and the division and the left01:57:29
right side by 5, all of this equals 301:57:32
point in time 3 seconds my point01:57:35
stopped. You can cross yourself01:57:38
the option is that you did everything at home01:57:41
a similar question will be about at what point01:57:45
a small time stopped here is important01:57:47
understand that if I stop, that's me01:57:50
the speed is zero so I have to find it01:57:51
the equation by which the speed is written01:57:53
equate to zero and find the instant of time01:57:57
it is possible to arise, it is possible to cross01:57:59
I don't know what else to do with champagne01:58:02
but you still want Well, here's a little Me01:58:05
it seems like everything is difficult, yes01:58:09
it's difficult, but don't worry at home01:58:13
home that it will be a little easier because I01:58:15
they tried for class today01:58:16
show the most difficult tasks01:58:18
take so that you can see that there is something more01:58:21
everything is more interesting at home01:58:23
of such a more adequate level, I think01:58:26
that it should be a little as simple as everything01:58:28
if there are any questions about01:58:30
I am waiting for you at home01:58:33
telegram if there are any questions about the lesson01:58:35
there are still unclear moments, please01:58:38
write to me in the program and I will explain more01:58:40
the third time, please write to me at01:58:42
Telegram, who needs it, I will tell you01:58:44
exhale everything Everything will be fine everything will be fine01:58:48
well, everything should be easy at home01:58:51
everything at home if there are any questions01:58:54
I'm waiting for personal ones01:58:57
enough, but it's okay to write01:58:59
We will deal with it individually01:59:00
chat today home Better Already01:59:03
don't do it better rest let it be you01:59:05
will get stuck in my head and there tomorrow morning01:59:08
open and go to everyone else something01:59:11
I need to explain, please throw it to me01:59:13
in Telegram, everything is homemade on the website01:59:18
your personal account is there01:59:19
button with home three equations01:59:23
and went to an additional lesson on this01:59:26
We will look at the situation if there is01:59:28
everything is really bad at home01:59:30
then we will think about something with you01:59:31
to do something This is not a problem at all01:59:33
see how it will be in your mood01:59:36
to go out At your home Everything is pa-pa to everyone01:59:38
I love everyone, I kiss everyone good nightDescription:
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