Let the function of two changes be given. We give the argument an increase, but the argument is too much invariable. The same function removes the increase, as it is called a private increase in change and is assigned:

Similarly, fixing the argument and giving the increment to the argument, we take away the private increase in the function behind the change:

The value is called the greatest increase in the function in points.

Appointment 4. The private function of two changeable ones is called between the change of the private increase of the function until the change of the given change, if the remaining value is zero (like the boundary). It is signified privately like this: either, or.

In this rank, for the appointed mayor:

Private func- tions are calculated according to the very rules and formulas, as if the function is one of change, it is protected from its own, which is differentiating according to change, it is important to be constant, and in case of differentiation it is important to change it.

Example 3. Know private fun functions:

Solution. a) In order to know the important constant value of that differential as a function of one variable:

Similarly, with respect to the constant value, we know:

Appointment 5. The total differential of a function is the sum of creations of private similar functions on the increase of independent independent ones, tobto.

Looking back at the fact that the differentials of independent changes are growing with their increments, that is. , the formula for the total differential can be written in the form

Example 4. Calculate the final differential of the function.

Solution. Oskіlki behind the formula of the total differential is known

Private holidays of the highest order

Private holidays are called private holidays of the first order or first private holidays.

Appointments 6. Private functions of a different order are called private functions of the first order.

Private chotiri of a different order. Vons are designated as follows:

Similarly, private losses of the 3rd, 4th and higher orders are assigned. For example, for the function may:

Private holidays of a different order, taken from different changes, are called changed private holidays. For function є pokhіdnі. It’s respectful that you’re in a mood, if you’re fluent without interruption, there’s room for jealousy.

Example 5. Change private functions in a different order

Solution. Private first order functions found in application 3:

Differentiation and change x and y, otrimaemo

Functions of two shifts, private shifts, differentials and gradient

Topic 5.Functions of two changes.

private holidays

Designated functions of two substitutions, methods of task.

Private holidays

Gradient function of one change

The value of the largest and the smallest value of the function of two variables in a closed area

1. Designated functions of a number of changes, ways of managing

For functions of two
the area of ​​appointment є deyak pointless point on the plane
, and the area of ​​value is the interval on the axis
.

For face-to-face appearance functions of two shifts zastosovutsya them lines.

Butt . For the function
induce a schedule and a line. Write down the line of line that passes through the point
.

Graph of linear functionsє flat in space.

For the graph function, the plane is to pass through the points
,
,
.

Lines of equal functionє parallel straight lines, equal
.

For linear functions of two variables
lines of equal are given to equals
і є family of parallel lines on the plane.

4

Function schedule 0 1 2 X

Lines of equal function

Private Proszvedeni functions of two

Let's look at the function
. Nadamo zminnoy at the point
fairly incremental
, overflowing the meaning of the change inevitable. Increased functionality

called private increase in the function of the change at the point
.

It is similarly indicated more private functionsby change: .

Appointmentprivate tour: , ,
,
.

Private free functions of the change is called the end of the border :

Designation: , ,
,
.

For the knowledge of private travel
behind the change are the rules of differentiation of the function of one change, vvazhuchi change postiynoy.

Similarly, for the knowledge of private hunting for a change change is respected .

Butt . For the function
know private trips
,
and calculate their values ​​at the point
.

Private outdoor functions
according to the change, you change the admission, which is fast:

We know the private, random functions, while respecting the fast ones:

Let's calculate the values ​​of private relatives at
,
:

;
.

Private walk in a different order The functions of a small number of changes are called private faucets in the first order.

Let's write for the 2nd order private behavior function:

;
;

;
.

;
and etc.

How to change the private functions of some of the changeable uninterrupted at the singing point
then stink equal to each other at this point. Also, for the function of two different values ​​of different similar private ones, do not lie in the order of differentiation:

.

butt. For the function to know private events in a different order
і
.

Solution

Zmishana is privately similar to the last differentiations on the cob function (vvazhayuchi fast), then differentiating like
(Respectfully fast).

Pokhіdna znahoditsya diferentiyuvannyam spochatku functions, then pokhіdnoi.

Zmіshanі privatnі pokhіdnі іvnі mіzh yourself:
.

3. Gradient of the function of two variables

Dominance of the gradient

Butt . given function
. Know the gradient
at the point
and try yoga.

Solution

We know the coordinates of the gradient - private slopes.

At the point
gradient goodbye. Cob vector
at the point, and end - at the point.

5

4. The value of the largest and the smallest value of the function of two variables in a closed area

Problem setting. Let's go on the square, the area is closed off
set by a system of irregularities mind
. It is necessary to know in the area of ​​the point, in which the function has the highest and the lowest value.

Important the task of knowing the extremum, a mathematical model for revenge linear obmezhennya (evenness, unevenness) that linear function
.

Problem setting. Find the highest and lowest value of a function
(2.1)

at dewatering

(2.2)

. (2.3)

Scales for the linear function in the middle regions
then the optimal solution, which delivers the goal function to the extreme, can only be reached at the cordon area. For the area specified by the line crossings, points of possible extremum є hotspots. Tse allows you to look at the breakdown of tasks graphic method.

Graphic display of the system of linear irregularities

For graphic development of this task, it is necessary to comprehend graphically the system of linear irregularities from two changes.

Order dіy:

It is significant that the nervousness
signifies right(view axis
), and unevenness
- upper coordinate line(view axis
).

butt. Virishity graphically nerіvnіst
.

We write down the alignment of the boundary line
and we will її behind two points, for example,
і
. Directly divide the area into two flats.

Point coordinates
satisfy the nervousness (
- verno), then, and the coordinates of all points on the surface, to avenge the point, satisfy the unevenness. The solution to the unevenness will be the coordinates of the points on the napіvploshchina, raztashovana prvoruch in the boundary straight line, including points on the cordon. Shukan on the surface of the little one was seen.

Solution
systems of irregularities are called admissible, which are non-negative coordinates , . The absence of admissible solutions to the system of irregularities sets up the area, as it is laid out in the first quarter of the coordinate plane.

butt. Induce the region of expansion of the system of irregularities

Razvyazannymi nerіvnosti є:

1)
- napіvploshchina, raztoshovana levoruch i lower vіdnosno straightї ( )
;

2)
- napіvploshchina, raztashovana in the right-lower pіvploschinі schodo straight line ( )
;

3)
- napіvploshchina, ruffled to the right of the straight line ( )
;

4) - napіvshchina higher axis abscissa, then straight ( )
.

0

Area of ​​acceptable solutions given a system of linear irregularities - the same pointless points, ruffled in the middle and on the border of the cutter
, what is peretina chotiriokh napіvploschin.

Geometric representation of a linear function

(lines of alignment and gradient)

The value is fixed
, we take equal
, which geometrically defines a straight line. At the skin point, the direct function gains significance і є the line of equivalence. Nadayuchi different meanings, for example,

, ... , we take the impersonal line equal - collection of parallel direct.

Let's gradient- Vector
coordinates of any equal to the values ​​of the coefficients with changing functions
. Danish vector: 1) perpendicular skin line (lines of line)
; 2) showing directly the growth of the target function.

Butt . Induce lines of alignment and gradient functions
.

The lines of the line at , , - are straight

,
,

, parallel one to one. Gradient is a vector, perpendicular to the skin line.

Graphical value of the largest and smallest value of the linear function in the area

Geometric statement of the problem. Find a point in the area of ​​rozv'yazkіv of the system of linear irregularities, which is to pass the line of line, which will give the largest (smallest) value of the linear function of the two changes.

Sequence dіy:

4. Find the coordinates of point A, violating the system of straight lines that overlap at point A, then calculate the least value of the function
. Similarly, for point B, the highest value of the function
. prompted for points. Privateget awayfunctions kіlkoh change that technique of differentiation. Extreme functionstwochange that yoga is necessary...

I continue to love the topic of mathematical analysis - let's go. At this article, we know private outdoor functions of the three: First Pokhіdnі and Other Pokhіdnі. What is necessary to know that in order to master the material? Do not believe it, ale, in a first way, it is necessary to remember the "primary" similar functions of one serpent - on a high chi I want to use the middle level. If it’s tight with them, then start learning the lesson How to know if I'm going to go? In another way, it’s important to read the article and comprehend-virishuvate, but not all, then most of the applications. Even though it’s already broken, then with a march we’ll go with me, if it’s cicavo, you’ll take away your satisfaction!

Methods and principles of knowledge three private functions really similar to the private func- tions of two different ones. The function of two changes, guessing, may look, de "iks" and "iplayer" - independent changes. Geometrically, the function of two substantives is the same surface in our trivial space.

The function of three changes can be seen, with which change they are called independentchange or arguments, change is called fallow or function. For example: - the function of three changes

And now a few words about fantastic films and aliens. Often you can feel about chotirivimirne, p'yatimirne, desyatimirne, etc. open spaces. What is it?
Even the function of the three change may take into account the fact that all the right things are in the chotirivimir space (really, the change chotiri). The schedule of the function of three zminnykh є so-called hypersurface. It is impossible to reveal її, oskolki we live in a trivi-worldly space (dovzhina / width / height). Sob you don’t get bored with me, I’m preaching a quiz. I'll put a power supply, and if you can, you can try the following on them:

- Chi is in the light of the fourth, p'yate thin. vimiryuvannya at the sense of upholstery rosemary space (dovzhina/width/height)?

- Can you please chotirivimirne, p'yativimirne then? open space for a wide roaming word? To bring an example of such space in our life.

- What can be more expensive in the past?

- What can be more expensive at the future?

- What are aliens?

On a be-yak basis, you can choose one of the following suggestions:
So / Ні (not fenced by science) / Not fenced by science / I don’t know

Whoever is correct in all food, the one who is better for everything, may be deak rich ;-)

Vіdpovіdі on zapitanya step by step I saw the hour of the lesson, do not miss the butt!

Alright, let's fly. I immediately good news: for the function of three changes, the rules of differentiation are valid and the table of similar. For that very reason, you need to be kind to the “superior” similar functions one change. Vіdmіnnosti zovsі not rich!

butt 1

Solution: It doesn't matter to guess - for the function of three zminnykh it is necessary three private similar ones of the first order, which are designated as follows:

Abo - private pokhіdna on "iks";
abo - private pokhіdna for the "iplayer";
abo - private pokhіdna "Z".

In the course of the move, there are more signs with a stroke, but the laying of collections, methods in the minds of tasks, you should also love victorious and cumbersome signs - so don’t be ruined! Possibly, not everyone knows how to correctly read out loud these terrible fractions. Butt: read like this: “de y po de iks”.

Let's look at the better for "iks": . If we know privately I will go to , then change і vvazhayutsya constants (constant numbers). And pokhіdna be-like constants, oh, grace, dovnyuє zero:

To show respect for the contracting index - you can’t defend anything for you, which are constants. So navit zruchnіshe, pochatkіvtsam I recommend vikoristovuvat itself such a record, less risk of getting lost.

(1) Vykoristuemo power of linearity is fickle, we blame all the constants for the sign of foulness. To give respect that another dodan does not need a constant of guilt: shards of “gravets” are a constant, then that is a constant. At the dodanka, for the ugly sign, there is an “equal” constant 8 and a constant “Z”.

(2) It is known the simplest pokhіdnі, not forgetting what the constants are. Dali zachіsuєmo vіdpovіd.

It's private. If we know I will privately go after the "iplayer", then change і respect constants:

(1) Vykoristovuemo domineering linearity. And again, respect that dodanki are constants, which means that nothing is needed as a sign of good guilt.

(2) It is well known, not forgetting, that the constants. Let's just say it.

I, nareshti, privately lost. If we know privately go to "Z", then change і respect constants:

Zagalne rule obvious and imperceptible: If we know privately I will gofor whatever independent change, thentwo others independent changes are valued by constants.

When filling out data, we will respect the following, but we will respect, zokrema, can't use contract indexes(how to indicate what kind of change to carry out differentiation). Entering the index will be a BAD FAILURE. Hmmm. it's funny, after such a zalyakuvannya I'll miss them here myself)

butt 2

Know the private behaviors of the first order of the functions of the three substantives

This is an example of an independent solution. Outwardly, the solution is that it is similar to the lesson.

Looked at two butts to do it easy and, having done a sprat of similar orders, to wind the teapot, try to deal with them orally.

Let's turn to the first meal of the quiz for the purpose of rallying: vimiryuvannya at the sense of upholstery rosemary space (dovzhina/width/height)?

Virna advice: Science is not fenced. All fundamental mathematical axiomatics, theorems, mathematical apparatus of the miraculous unsuperbly practice at the expanse, whether it be rozmirnosti. It is not included, that here in the All-World there is a hypersurface that is indispensable to our mind, for example, a hypersurface, as the function of the three zminnyh is set. And maybe hyper-surface entrusted to us, or to inspire us directly in them, just our zir, other organs are more sensitive, svіdomіst zdatnі to spriynyattya that comprehension is less than three vimirіv.

Let's turn to applications. So, if you are very interested in the quiz, it’s better to read it on your feet if you learn how to know the private func- tions of three of them, otherwise I’ll blame the whole brain for you in the course of the article =)

Crim the simplest applications 1,2 in practice, the tasks are being worked out, as if they could be called a small puzzle. So apply, to my annoyance, they wiped the dawn from the field, if I made a lesson Private outdoor functions of two people. We must have wasted:

butt 3

Solution: yak bi here "everything is simple", but first of all, the anger is tempting. When you know private, rich people, someone will tell fortunes in the thick of the forest and have mercy.

Let's look at the butt sequentially, clearly and consciously.

Pochnemo s private pokhіdnoї s "iks". If we know privately I will go to "iks", then change them with constants. Otzhe, the indicator of our function is also a constant. For teapots, I recommend an offensive solution: in black, remember the constant on a specific number, a positive number, for example, on “five”. As a result, we will see the function of one change:
otherwise you can write it like this:

Tse static function with a folding basis (sine). By:

Now let's guess, scho, in this order:

On a clean copy, obviously, the decision should be drawn up like this:

We know that I will privately go after the "iplayer", they are respected by constants. If "iks" is a constant, then tezh is a constant. On blacknets, the same trick is tried: replace, for example, by 3, “Z” - replace by the same “five”. As a result, the function of one change will reappear:

Tse showing function with folding indicator. per the rule of differentiation of the collapsible function:

Now let's make our change:

In this manner:

On a clean copy, I realized, the design can look good:

І mirror-like drop from a private similar to “z” (-constant):

For the singing dosvіdu carrying out analіz it is possible to carry out thoughts.

Let's take another part of the task - we fold the differential of the first order. It is even simpler, for analogy with the function of two variables, the differential of the first order is written with the following formula:

In this view:

I'm sorry. I will designate that in practical tasks the first-order differential of the functions of the three variables should be added significantly more similarly, the lower functions of the two variables.

A funny butt for an independent cherry:

butt 4

Find the private first order differentials of the function of three variables and add the final differential of the first order

Outwardly, the solution is that it is similar to the lesson. To blame for the difficulties, vindicate the "Chainikov's" algorithm, you can be guaranteed to help. І sche korisna porada - don't hurry. I don't mind such examples.

Let's take a look at another food: Can you please chotirivimirne, p'yativimirne toshcho? open space for a wide roaming word? To bring an example of such space in our life.

Virna advice: So. And it's easy too. For example, dodaemo to the length/width/height of the fourth wimir - an hour. The popular chotirivimirny expanse-chas and all vіdoma the theory of viability, neatly stolen by Einstein from Lobachevsky, Poincari, Lorenz and Minkovsky. You don't know everything. Why did Einstein get the Nobel Prize? The science world has a terrible scandal, and the Nobel Committee, having formulated the merit of the plagiarist approximately as follows: "For the high contribution to the development of physics." So out. The brand of Einstein's trio is pure promotion and PR.

It is easy to add five vimir to the open space, for example: an atmospheric vice. And so far away, so far away, so far away, put the scallops on your model - the stiles will be. We live in a wide meaning of the word in a rich and wide space.

Let's take a couple of typical tasks:

butt 5

Know private events first order at the point

Solution: The task of such a formula is often used in practice, that is, the transmission of the coming two days:
- It is necessary to know the private events of the first order;
- It is necessary to designate the values ​​of private relatives of the 1st order in points.

We see:

(1) Before us is a collapsible function, and on the first line, take a similar arc tangent. In this case, in fact, I’ll use the tabular formula of the similar arc tangent. per the rule of differentiation of the collapsible function The result must be multiplied by the appropriate internal function (nesting): .

(2) Vykoristovuemo domineering linearity.

(3) I take it easy, what is lost, not forgetting, what are constants.

It is necessary to know the meaning of the found private value at the point for the mind's task. Let's assume the coordinates of the point y are known to be lost:

The advantage of this task is the fact that other private parties are known for a similar scheme:

Yak bachite, the pattern of virishenya is practically the same.

Let us calculate the value of the found private value in points:

І, nareshti, similar to "Z":

Ready. The solution can also be filled in another way: first, you need to know all three private dates, and then calculate their values ​​at the point. Ale, I’m guessing, guidance is a good way - only they knew privately, and right away, without looking at the casi, they lied about the meaning of the point.

It means that a geometrical point is a fully real point of our trivial space. The meaning of the function, similar ones - already the fourth world, and definitively geometrically known, no one knows. As it seems, I didn’t call anyone with a roulette wheel, without perverting it.

If the philosophical theme has come up again, let's look at the third food: What can be more expensive in the past?

Virna advice: Hi. It will be more expensive in the past to supervise another law of thermodynamics about the non-reversibility of physical processes (entropy). So don’t pirnayte, be kind, into the pool without water, you can turn it back only at the video recording =) Folk wisdom did not for nothing guess the life of the law: “Sim once in the world, once in the air.” Wanting, really a sumptuous thing, the hour of one-way directing and non-return, none of us will be younger tomorrow. And different fantastic films on the “Terminator” kshtalt from a scientific point of view are a tsіlkovita nіsenіtnitsa. Absurdity and philosophy's glance - if the Effect, turning at the past, can destroy the power of the Cause. .

Tsіkavіshe z pokhіdnoy on “zet”, wanting, all the same may be the same:

(1) We blame the constant for the sign of the worse.

(2) Here I am re-documenting two functions, skin vіd "live" change "z". In principle, you can work out the formula of a similarly private one, but it’s easier to follow a different path - to know the best way to work.

(3) Pokhіdna - tse tabular pokhіdna. The other dodanka already knows the foldable function.

butt 9

Know the private behaviors of the first order of the functions of the three substantives

This is an example of an independent solution. Think how rationally you know that chi іnsha privately go. Outwardly, the solution is that it is similar to the lesson.

Before that, go to the final examples of the lesson and take a look private trips in a different order functions of the three replacements, all again for the fourth power:

Qi can be more expensive at the future?

Virna advice: Science is not fenced. Paradoxically, but there is no mathematical, physical, chemical, or other natural law, which has hindered the future more dearly! Are you a nіsenіtnitse? But it’s practical for the skin in life to change (and, moreover, not supported by any logical arguments), what will happen that chi іnsha podіya. And out of the blue! Did you get the information? From the future? In this rank, fantastic films about the future are more expensive, that one, before the speech, the transfer of all the powers that be, psychics cannot be called such a marenny. Accept science who did not catch. Everything is possible! So, if I studied at school, then CDs and flat screen monitors from films were created less like fantastic fiction.

Vіdoma comedy "Ivan Vasilyovich changing his profession" is half a guess (like a maximum). The current scientific law did not prevent Ivan the Terrible from opinating in the future, but it is impossible that two peppers would opine in the past and beat the tsar's bindings.

Let's look at the function in two ways:

Shards of change $x$ and $y$ are independent, for such a function it is possible to provide an understanding of private information:

Private function $f$ at point $M=\left(((x)_(0));((y)_(0)) \right)$ for change $x$ -

\[(((f)")_(x))=\underset(\Delta x\to 0)(\mathop(\lim ))\,\frac(f\left(((x)_(0) )+\Delta x;((y)_(0)) \right))(\Delta x)\]

In the same way, you can assign a private fee for a change of $y$:

\[(((f)")_(y))=\underset(\Delta y\to 0)(\mathop(\lim ))\,\frac(f\left(((x)_(0) );((y)_(0))+\Delta y \right))(\Delta y)\]

In other words, in order to know the private func- tions of some of the change, it is necessary to fix the decision of the change, the crime of shukano, and then we will know the zvichayna to go after the price of the change.

Sounds like the main trick for counting such lousy ones: just take into account that everything is changing, krym tsієї, є constant, after which differentiate the function so that you would differentiate the “singular” - with one zminnoy. For example:

$\begin(align)& ((\left(((x)^(2))+10xy \right))_(x))^(\prime )=((\left(((x)^(2 ) )) \right))^(\prime ))_(x)+10y\cdot ((\left(x \right))^(\prime ))_(x)=2x+10y, \\& ( ( \left(((x)^(2))+10xy \right))_(y))^(\prime )=((\left(((x)^(2)) \right))^( \ prime ))_(y)+10x\cdot ((\left(y \right))^(\prime ))_(y)=0+10x=10x. \\\end(align)$

Obviously, it is normal to give private holidays from different changes. Why is it more important to understand, why, let's say, in the first one we were calmly charged $10y$ s-pid of a bad sign, and in the other - the first one was zeroed out. Everything is conceived through those that all letters, krіm zminnoi, for some kind of differentiation, are respected by constants: they can be blamed, spat, etc.

What is "private fun"?

Today we will talk about the functions of a few changers and about private holidays in them. First of all, what is the function of a few replacements? Dosi mi called to change the function like $y\left(x \right)$ or $t\left(x \right)$, otherwise change that one-single function in it. Now there will be only one function in us, and there will be a change of sprat. If you change $y$ and $x$ the value of the function will change. For example, if $x$ increases twice, the value of the function changes, if $x$ changes, but $y$ does not change, the value of the function changes itself.

It was understood that the function in the form of a number of variables, just like in one of the variables, can be differentiated. However, the oskіlki zmіnnykh kіlka, then it is possible to differentiate from different zmіnnyh. For whom, specific rules are blamed, which are the same when differentiating one change.

First for everything, if we want to lose our functions, if we are somehow changeable, then we are to blame, for what kind of change we are supposed to leave - that’s why it’s called a private mess. For example, we have a function in the form of two substitutions, and we can frighten її like $x$, so $y$ — two private ones similar to the skin of the interchangeable ones.

In a different way, if we have fixed one of the changes and we start to respect privately after it, then everything else that enters the function is respected by constants. For example, $z\left(xy \right)$, as we are important to privately walk around $x$, then, squinting, demi-simply $y$, we are important to be a constant and to be treated by itself as a constant. Zakrema, when counting bad things, we can blame $y$ for the shackle (we have a constant), but when counting bad money, as we have here, it’s like a virus to avenge $y$ and not avenge $x$, then it’s good virazu dorivnyuvatime "zero" like a good constant.

At first glance, you can get away that I tell you about it in a folded way, and a lot of learners stray on the cob. There is nothing supernatural among the private ones, and we are changing from the butt of specific tasks.

Responsible for radicals and rich members

Manager No. 1

Sob not to waste an hour, from the very cob we’ll start with serious butts.

For starters, I guess the following formula:

This is the standard table value, as we know from the standard course.

It's good for someone to use $z$ like this:

\[(((z)")_(x))=((\left(\sqrt(\frac(y)(x)) \right))^(\prime ))_(x)=\frac( 1)(2\sqrt(\frac(y)(x)))((\left(\frac(y)(x) \right))^(\prime ))_(x)\]

Let's once again, the shards under the roots cost not $x$, but some other viraz, in this case $\frac(y)(x)$, then we speed up the standard tabular values, and then, the shards under the roots cost not $x $, and another viraz, it is necessary for us to multiply our expenses for one more viraz for the other viraz. Let's start stepping on the cob:

\[((\left(\frac(y)(x) \right))^(\prime ))_(x)=\frac((((((y)"))_(x))\cdot xy \cdot ((((x)"))_(x)))(((x)^(2)))=\frac(0\cdot xy\cdot 1)(((x)^(2) ) )=-\frac(y)(((x)^(2)))\]

Let's turn to our virazu and write down:

\[(((z)")_(x))=((\left(\sqrt(\frac(y)(x)) \right))^(\prime ))_(x)=\frac( 1)(2\sqrt(\frac(y)(x)))((\left(\frac(y)(x) \right))^(\prime ))_(x)=\frac(1) (2\sqrt(\frac(y)(x)))\cdot \left(-\frac(y)(((x)^(2))) \right)\]

Everything is in principle. However, it is wrong to leave її in such a look: it’s not handy to beat such a construction for the distant ones, so let’s do it a trifle:

\[\frac(1)(2\sqrt(\frac(y)(x)))\cdot \left(-\frac(y)(((x)^(2))) \right)=\frac (1)(2)\cdot \sqrt(\frac(x)(y))\cdot \frac(y)(((x)^(2)))=\]

\[=-\frac(1)(2)\cdot \sqrt(\frac(x)(y))\cdot \sqrt(\frac(((y)^(2)))(((x)^ (4))))=-\frac(1)(2)\sqrt(\frac(x\cdot ((y)^(2)))(y\cdot ((x)^(4)))) =-\frac(1)(2)\sqrt(\frac(y)(((x)^(3))))\]

Vidpovid found. Now let's deal with $y$:

\[(((z)")_(y))=((\left(\sqrt(\frac(y)(x)) \right))^(\prime ))_(y)=\frac( 1)(2\sqrt(\frac(y)(x)))\cdot ((\left(\frac(y)(x) \right))^(\prime ))_(y)\]

Vipishemo okremo:

\[((\left(\frac(y)(x) \right))^(\prime ))_(y)=\frac((((((y)"))_(y))\cdot xy \cdot ((((x)"))_(y)))(((x)^(2)))=\frac(1\cdot xy\cdot 0)(((x)^(2) ) )=\frac(1)(x)\]

Now we write:

\[(((z)")_(y))=((\left(\sqrt(\frac(y)(x)) \right))^(\prime ))_(y)=\frac( 1)(2\sqrt(\frac(y)(x)))\cdot ((\left(\frac(y)(x) \right))^(\prime ))_(y)=\frac( 1)(2\sqrt(\frac(y)(x)))\cdot \frac(1)(x)=\]

\[=\frac(1)(2)\cdot \sqrt(\frac(x)(y))\cdot \sqrt(\frac(1)(((x)^(2))))=\frac (1)(2)\sqrt(\frac(x)(y\cdot ((x)^(2))))=\frac(1)(2\sqrt(xy))\]

Everything is shattered.

Manager No. 2

This butt is at once simpler and more folding, lower forward. More folded, to that there is more action here, but simpler, to that there is no root here, moreover, the function is symmetrical to $x$ and $y$, tobto. As we remember $x$ and $y$ as missions, the formula does not seem to change. Tse respect had to be forgiven for the payment of private expenses, tobto. It's enough to damage one of them, and in the other one just remember $x$ and $y$ with the brushes.

Let's get to the point:

\[(((z)")_(x))=((\left(\frac(xy))(((x)^(2))+((y)^(2))+1) \ right ))^(\prime ))_(x)=\frac(((\left(xy \right))^(\prime ))_(x)\left(((x)^(2))+ ( (y)^(2))+1 \right)-xy((\left(((x)^(2))+((y)^(2))+1 \right))^(\prime ) )_(x))(((\left(((x)^(2))+((y)^(2))+1 \right))^(2)))\]

Let's get excited:

\[((\left(xy \right))^(\prime ))_(x)=y\cdot ((\left(x \right))^(\prime ))=y\cdot 1=y\ ]

Prote richly learn such a record of ignorance, we will write down the axis like this:

\[((\left(xy \right))^(\prime ))_(x)=((\left(x \right))^(\prime ))_(x)\cdot y+x\cdot ((\left(y \right))^(\prime ))_(x)=1\cdot y+x\cdot 0=y\]

In this rank, we once again switch over to the universality of the algorithm of private relatives: they didn’t care about them, if all the rules are set up correctly, you will be the one yourself.

Now let's take a look at one more private trick of our great formula:

\[((\left(((x)^(2))+((y)^(2))+1 \right))^(\prime ))_(x)=((\left((( x)^(2)) \right))^(\prime ))_(x)+((\left(((y)^(2)) \right))^(\prime ))_(x) +(((1)")_(x))=2x+0+0\]

Let's assume that we take away the dependence on our formula and take it away:

\[\frac(((\left(xy \right))^(\prime ))_(x)\left(((x)^(2))+((y)^(2))+1 \ right)-xy((\left(((x)^(2))+((y)^(2))+1 \right))^(\prime ))_(x))(((\left (((x)^(2))+((y)^(2))+1 \right))^(2)))=\]

\[=\frac(y\cdot \left(((x)^(2))+((y)^(2))+1 \right)-xy\cdot 2x)(((\left((( ( x)^(2))+((y)^(2))+1 \right))^(2)))=\]

\[=\frac(y\left(((x)^(2))+((y)^(2))+1-2((x)^(2)) \right))(((\ left(((x)^(2))+((y)^(2))+1 \right))^(2)))=\frac(y\left(((y)^(2)) -((x)^(2))+1 \right))(((\left(((x)^(2))+((y)^(2))+1 \right))^(2 )))\]

$x$ is reinstated. And in order to fix $y$ in the same viraz, let's not vikonuvat all the same sequence of diy, but rather with the symmetry of our vivid viraz - we simply replace in our vivid viraz all $y$ by $x$ and navpak:

\[(((z)")_(y))=\frac(x\left(((x)^(2))-((y)^(2))+1 \right))((( ( \left(((x)^(2))+((y)^(2))+1 \right))^(2)))\]

For the rahunok of symmetry, they praised the whole viraz richly shvidshe.

nuance cherry

For the private ones, all standard formulas are used, which is the best for the private ones, but the same is true for the private ones. With this, however, they blame their own specific features: if we respect $x$ privately, then if we take її for $x$, then we consider it as a constant, and to that її is similar to more expensive “zero”.

Like and at the same time with the most significant pokhіdnymi, private (one and the same) you can spoil a kіlkom in different ways. For example, the same construction, which was so well applauded, can be rewritten like this:

\[((\left(\frac(y)(x) \right))^(\prime ))_(x)=y\cdot ((\left(\frac(1)(x) \right)) ^(\prime ))_(x)=-y\frac(1)(((x)^(2)))\]

\[((\left(xy \right))^(\prime ))_(x)=y\cdot (((x)")_(x))=y\cdot 1=y\]

At once about those, from the other side, you can beat the formula in the form of a casual sum. As we know, there are more expensive sums of the dead. For example, let's write this:

\[((\left(((x)^(2))+((y)^(2))+1 \right))^(\prime ))_(x)=2x+0+0=2x \]

Now, knowing everything, let's try to improve with more serious usages, the shards of right private tricks are not surrounded by only rich terms and roots: trigonometry, logarithms, and display functions are used there. Now let's get busy.

Task with trigonometric functions and logarithms

Manager No. 1

We write the following standard formulas:

\[((\left(\sqrt(x) \right))^(\prime ))_(x)=\frac(1)(2\sqrt(x))\]

\[((\left(\cos x \right))^(\prime ))_(x)=-\sin x\]

Having mastered this knowledge, let's try to verse:

\[(((z)")_(x))=((\left(\sqrt(x)\cdot \cos \frac(x)(y) \right))^(\prime ))_(x )=((\left(\sqrt(x) \right))^(\prime ))_(x)\cdot \cos \frac(x)(y)+\sqrt(x)\cdot ((\left (\cos \frac(x)(y) \right))^(\prime ))_(x)=\]

Okremo write one change:

\[((\left(\cos \frac(x)(y) \right))^(\prime ))_(x)=-\sin \frac(x)(y)\cdot ((\left( \frac(x)(y) \right))^(\prime ))_(x)=-\frac(1)(y)\cdot \sin \frac(x)(y)\]

Turn to our design:

\[=\frac(1)(2\sqrt(x))\cdot \cos \frac(x)(y)+\sqrt(x)\cdot \left(-\frac(1)(y)\cdot \sin \frac(x)(y) \right)=\frac(1)(2\sqrt(x))\cdot \cos \frac(x)(y)-\frac(\sqrt(x))( y)\cdot \sin \frac(x)(y)\]

We all know about $x$, now let's get down to calculating $y$:

\[(((z)")_(y))=((\left(\sqrt(x)\cdot \cos \frac(x)(y) \right))^(\prime ))_(y )=((\left(\sqrt(x) \right))^(\prime ))_(y)\cdot \cos \frac(x)(y)+\sqrt(x)\cdot ((\left (\cos \frac(x)(y) \right))^(\prime ))_(y)=\]

Well, I know, I’m afraid one viraz:

\[((\left(\cos \frac(x)(y) \right))^(\prime ))_(y)=-\sin \frac(x)(y)\cdot ((\left( \frac(x)(y) \right))^(\prime ))_(y)=-\sin \frac(x)(y)\cdot x\cdot \left(-\frac(1)(( (y)^(2))) \right)\]

Let's turn to the end of the day and continue to see:

\[=0\cdot \cos \frac(x)(y)+\sqrt(x)\cdot \frac(x)(((y)^(2)))\sin \frac(x)(y) =\frac(x\sqrt(x))(((y)^(2)))\cdot \sin \frac(x)(y)\]

Everything is shattered.

Manager No. 2

Let's write down the formula we need:

\[((\left(\ln x \right))^(\prime ))_(x)=\frac(1)(x)\]

Now I'm sorry for $x$:

\[(((z)")_(x))=((\left(\ln \left(x+\ln y \right) \right))^(\prime ))_(x)=\frac( 1)(x+\ln y).((\left(x+\ln y \right))^(\prime ))_(x)=\]

\[=\frac(1)(x+\ln y)\cdot \left(1+0 \right)=\frac(1)(x+\ln y)\]

Found for $x$. Important for $y$:

\[(((z)")_(y))=((\left(\ln \left(x+\ln y \right) \right))^(\prime ))_(y)=\frac( 1)(x+\ln y).((\left(x+\ln y \right))^(\prime ))_(y)=\]

\[=\frac(1)(x+\ln y)\left(0+\frac(1)(y) \right)=\frac(1)(y\left(x+\ln y \right))\ ]

The task is over.

nuance cherry

Later, since the functions were not taken privately, the rules are overwritten by the same ones, regardless of whether they work with trigonometry, with roots or with logarithms.

The classic rules of work are always replaced by the standard ones, and at the same time, the sum of the retail, private and collapsible functions.

The rest of the formula is most often explained at the end of the day when the meeting is over with private holidays. Mi zustrіchaєmosya with them practically skrіz. There hasn’t yet been a city manager, so that we don’t get out there. But if we didn’t squirm with the formula, we still get one more benefit, and for ourselves, the peculiarity of the work with private walks. So we fix one change, the lines are constants. Zocrema, as we respect the privately lost virase $\cos \frac(x)(y)$ $y$, then $y$ itself is changed, and $x$ is overwritten with a constant. The same practice and navpaki. Її can be blamed for the bad sign, but bad as the constant itself is more like “zero”.

Everything should be brought to the point that private looks of one and the same viraz, but from different changes they can look differently. For example, marveling at such a virazi:

\[((\left(x+\ln y \right))^(\prime ))_(x)=1+0=1\]

\[((\left(x+\ln y \right))^(\prime ))_(y)=0+\frac(1)(y)=\frac(1)(y)\]

Task with demonstrative functions and logarithms

Manager No. 1

Let's write down the following formula:

\[((\left(((e)^(x)) \right))^(\prime ))_(x)=((e)^(x))\]

Knowing this fact, as well as the foldable functions, we can try to frighten. I believe in two different ways at once. The first and most obvious is the cost of the work:

\[(((z)")_(x))=((\left(((e)^(x))\cdot ((e)^(\frac(x)(y))) \right) )^(\prime ))_(x)=((\left(((e)^(x)) \right))^(\prime ))_(x)\cdot ((e)^(\frac (x)(y)))+((e)^(x))\cdot ((\left(((e)^(\frac(x)(y))) \right))^(\prime ) )_(x)=\]

\[=((e)^(x))\cdot ((e)^(\frac(x)(y)))+((e)^(x))\cdot ((e)^(\frac ) (x)(y)))\cdot ((\left(\frac(x)(y) \right))^(\prime ))_(x)=\]

Let's see this viraz:

\[((\left(\frac(x)(y) \right))^(\prime ))_(x)=\frac(((((x)"))_(x))\cdot yx .(((((y)"))_(x)))(((y)^(2)))=\frac(1\cdot yx\cdot 0)((((y)^(2) )) =\frac(y)((((y)^(2)))=\frac(1)(y)\]

Let's turn to our design and continue to see it:

\[=((e)^(x))\cdot ((e)^(\frac(x)(y)))+((e)^(x))\cdot ((e)^(\frac ) (x)(y)))\cdot \frac(1)(y)=((e)^(x))\cdot ((e)^(\frac(x)(y)))\left( 1 +\frac(1)(y)\right)\]

Everything, $x$ is covered.

However, as I said, at the same time we will try to protect my privacy in a different way. For whom respectfully so:

\[((e)^(x))\cdot ((e)^(\frac(x)(y)))=((e)^(x+\frac(x)(y)))\]

We write it down like this:

\[((\left(((e)^(x))\cdot ((e)^(\frac(x)(y))) \right))^(\prime ))_(x)=( (\left(((e)^(x+\frac(x)(y))) \right))^(\prime ))_(x)=((e)^(x+\frac(x)(y ) )))\cdot ((\left(x+\frac(x)(y) \right))^(\prime ))_(x)=((e)^(x+\frac(x)(y) ) )\cdot \left(1+\frac(1)(y) \right)\]

As a result, we took away the same amount of money, and the prote was charged as the smaller one. For whom to finish the bulk remember that when you finish the show, you can add up.

Now I'm sorry for $y$:

\[(((z)")_(y))=((\left(((e)^(x))\cdot ((e)^(\frac(x)(y))) \right) )^(\prime ))_(y)=((\left(((e)^(x)) \right))^(\prime ))_(y)\cdot ((e)^(\frac (x)(y)))+((e)^(x))\cdot ((\left(((e)^(\frac(x)(y))) \right))^(\prime ) )_(y)=\]

\[=0\cdot ((e)^(\frac(x)(y)))+((e)^(x))\cdot ((e)^(\frac(x)(y))) \cdot ((\left(\frac(x)(y) \right))^(\prime ))_(y)=\]

Let's sing one viraz okremo:

\[((\left(\frac(x)(y) \right))^(\prime ))_(y)=\frac(((((x)"))_(y))\cdot yx \cdot ((((y)"))_(y)))(((y)^(2)))=\frac(0-x\cdot 1)(((y)^(2))) =-\frac(1)((((y)^(2)))=-\frac(x)(((y)^(2)))\]

We sell the version of our external design:

\[=((e)^(x))\cdot ((e)^(\frac(x)(y)))\cdot \left(-\frac(x)(((y)^(2) )) \right)=-\frac(x)(((y)^(2)))\cdot ((e)^(x))\cdot ((e)^(\frac(x)(y) ))\]

It dawned on me that I could have lost my way in another way, I would have looked like this myself.

Manager No. 2

Fuck for $x$:

\[(((z)")_(x))=((\left(x \right))_(x))\cdot \ln \left(((x)^(2))+y \right )+x\cdot ((\left(\ln \left(((x)^(2))+y \right) \right))^(\prime ))_(x)=\]

Let's stop one viraz okremo:

\[((\left(\ln \left(((x)^(2))+y \right) \right))^(\prime ))_(x)=\frac(1)(((( x )^(2))+y)\cdot ((\left(((x)^(2))+y \right))^(\prime ))_(x)=\frac(2x)(( ((x)^(2))+y)\]

Sold solution of exterior design: $$

The axis is so clear.

Lost for analogy to know by $y$:

\[(((z)")_(y))=((\left(x \right))^(\prime ))_(y).\ln \left(((x)^(2)) +y \right)+x\cdot ((\left(\ln \left(((x)^(2))+y \right) \right))^(\prime ))_(y)=\]

One viraz, it’s ok, like a zavzhdi okremo:

\[((\left(((x)^(2))+y \right))^(\prime ))_(y)=((\left(((x)^(2)) \right) )^(\prime ))_(y)+(((y)")_(y))=0+1=1\]

Prodovzhuєmo virіshennya main designії:

Everything is covered. Like a bachite, fallow, depending on how the change is taken for differentiation, they appear absolutely different.

nuance cherry

The axis of the yaskra is an example of how one and the same functions can be damaged in two different ways. Axis to wonder:

\[(((z)")_(x))=\left(((e)^(x))\cdot ((e)^(\frac(x)(y))) \right)=( (\left(((e)^(x)) \right))^(\prime ))_(x)\cdot ((e)^(\frac(x)(y)))+((e) ^(x))\cdot ((\left(((e)^(\frac(x)(y))) \right))^(\prime ))_(x)=\]

\[=((e)^(x))\cdot ((e)^(\frac(x)(y)))+((e)^(x))\cdot ((e)^(\frac ) (x)(y)))\cdot \frac(1)(y)=((e)^(x))\cdot ((e)^(^(\frac(x)(y)))) )\ left(1+\frac(1)(y) \right)\]

\[(((z)")_(x))=((\left(((e)^(x)).((e)^(\frac(x)(y))) \right)) ^(\prime ))_(x)=((\left(((e)^(x+\frac(x)(y))) \right))^(\prime ))_(x)=(( e)^(x+\frac(x)(y))).((\left(x+\frac(x)(y) \right))^(\prime ))_(x)=\]

\[=((e)^(x))\cdot ((e)^(^(\frac(x)(y))))\left(1+\frac(1)(y) \right)\ ]

When choosing different paths, the calculation could be different, but if it’s true, it’s all right, you see it yourself. Prices are worthy of the classical, and private of the later ones. I’ll guess again from whom: it’s fallow, it’s like, what a change, I’ll take a good one, that’s it. differentiation, vіdpovіd can vyyti zovsіm raznoyu. Marvel:

\[((\left(\ln \left(((x)^(2))+y \right) \right))^(\prime ))_(x)=\frac(1)(((( x )^(2))+y)\cdot ((\left(((x)^(2))+y \right))^(\prime ))_(x)=\frac(1)(( (( x)^(2))+y)\cdot 2x\]

\[((\left(\ln \left(((x)^(2))+y \right) \right))^(\prime ))_(y)=\frac(1)(((( x )^(2))+y)\cdot ((\left(((x)^(2))+y \right))^(\prime ))_(y)=\frac(1)(( ((x)^(2))+y)\cdot 1\]

Nasamkinets for fixing all the material, let's try to fix two butts.

Task with a trigonometric function and a function with three changes

Manager No. 1

Let's write these formulas:

\[((\left(((a)^(x)) \right))^(\prime ))=((a)^(x))\cdot \ln a\]

\[((\left(((e)^(x)) \right))^(\prime ))=((e)^(x))\]

Let's now virishuvate our viraz:

\[(((z)")_(x))=((\left(((3)^(x\sin y)) \right))^(\prime ))_(x)=((3 )^(x.\sin y))\cdot \ln 3\cdot ((\left(x\cdot \sin y \right))^(\prime ))_(x)=\]

Okremo porahuemo such a design:

\[((\left(x\cdot \sin y \right))^(\prime ))_(x)=(((x)")_(x))\cdot \sin y+x((\ left(\sin y \right))^(\prime ))_(x)=1\cdot \sin y+x\cdot 0=\sin y\]

Prodovzhuєmo virishuvati vihіdny viraz:

\[=((3)^(x\sin y))\cdot \ln 3\cdot \sin y\]

This is the residual amount of private change $x$. Now I'm sorry for $y$:

\[(((z)")_(y))=((\left(((3)^(x\sin y)) \right))^(\prime ))_(y)=((3 )^(x\sin y))\cdot \ln 3\cdot ((\left(x\sin y \right))^(\prime ))_(y)=\]

Virishimo one viraz okremo:

\[((\left(x\cdot \sin y \right))^(\prime ))_(y)=(((x)")_(y))\cdot \sin y+x((\ left(\sin y \right))^(\prime ))_(y)=0\cdot \sin y+x\cdot \cos y=x\cdot \cos y\]

Virishuemo to the end of our design:

\[=((3)^(x\cdot \sin y))\cdot \ln 3\cdot x\cos y\]

Manager No. 2

At first glance, this butt can be folded, because there are three changes. Indeed, it is one of the simplest tasks for today's video tour.

Known by $x$:

\[(((t)")_(x))=((\left(x((e)^(y))+y((e)^(z)) \right))^(\prime ) )_(x)=((\left(x\cdot ((e)^(y)) \right))^(\prime ))_(x)+((\left(y\cdot ((e) ) ^(z)) \right))^(\prime ))_(x)=\]

\[=((\left(x \right))^(\prime ))_(x)\cdot ((e)^(y))+x\cdot ((\left(((e)^(y ) )) \right))^(\prime ))_(x)=1\cdot ((e)^(y))+x\cdot o=((e)^(y))\]

Now let's look at $y$:

\[(((t)")_(y))=((\left(x\cdot ((e)^(y))+y\cdot ((e)^(z)) \right))^ (\prime ))_(y)=((\left(x\cdot ((e)^(y)) \right))^(\prime ))_(y)+((\left(y\cdot ) ((e)^(z)) \right))^(\prime ))_(y)=\]

\[=x\cdot ((\left(((e)^(y)) \right))^(\prime ))_(y)+((e)^(z))\cdot ((\left (y \right))^(\prime ))_(y)=x\cdot ((e)^(y))+((e)^(z))\]

We knew the truth.

Now it's too much to know $z$:

\[(((t)")_(z))=((\left(x\cdot ((e)^(y))+((y)^(z)) \right))^(\prime ))_(z)=((\left(x\cdot ((e)^(y)) \right))^(\prime ))_(z)+((\left(y\cdot ((e )^(z)) \right))^(\prime ))_(z)=0+y\cdot ((\left(((e)^(z)) \right))^(\prime )) _(z)=y\cdot ((e)^(z))\]

We praised the third pokhidna, on which the vision of another task is completed again.

nuance cherry

Like a bachite, there is nothing folding in these two butts. The only thing, why we messed up, it’s in the fact that the folding functions are often stagnant and fallow, in addition, it’s privately funny, we’ll have to change depending on the situation.

In the rest of the task, we were asked to work out the functions of three different ones. There is nothing terrible in tsomu, prote naprikintsі mi have changed, that all the stinks are one in the same day.

Key moments

The rest of the vysnovki from today's video lesson is as follows:

1. Private expenses are taken into account as such, as if they were important, in order to take into account private expenses by one change, deciding all the changes that are included in this function, we take them as constants.
2. Pratsyyuyuchi s private pokhіdnymi vikoristovuєmo tі і themselves standard formulas, like і z the most important pokhіdnymi: sum, raznitsyu, pokhіdnu create і private і, zrozumіlo, pokhіdnu foldable functions.

Obviously, reviewing one video lesson is not enough, so that I can expand on these topics, so right now on my site, I have a set of tasks dedicated to this day's topics right now on my site - come in, zavantazhyte, vypishuyte tsі zavdannya and zvіryayyaytes. After all, you won’t have any everyday problems from private ones like sleeping or working independently. Obviously, this is far from the last lesson in modern mathematics, so go to our website, add VKontakte, subscribe to YouTube, put likes and follow us!

Private casual functions of some of the changeables are functions of the changers themselves. These functions, in their own right, can be mothers of private functions, as they are called by other private functions (or private of a different order) external functions.

So, for example, the function of two alternating maє chotir privately in a different order, as they signify and are designated by the coming rank:

The function of three changes may be nine private similar ones in a different order:

In a similar way, the private names of the third and highest order of the function of the number of changes are designated and designated: the private order of the function of the number of changes is called the private order of the first order in the private order of the same function.

For example, a private function of the third order is a private function of the first order in a private similar of another order

It is a private waste of a different order, taken for a dekilkom by various changes, it is called a mixed private waste.

For example, private holidays

є zm_shanimi private similar functions of two zminnyh.

butt. Know the changes in private functions in a different order

Solution. We know private trips of the first order

Then we know about changing private events in a different order

Mi, scho zmіshanі privatnі vіdnі і vіdmіnі mіzh yоmself іѕ аn order of differentiation, i.e. posіdіvnіstyu, yakіy viroblyаієі dіfіentіvіvannі z rіznіh zmіnnim, yоu appeared tоtоtоno equal. Tsey result nevipadkovy. Wherever there are private similar cases, such a theorem is accepted without proof.