Partial derivatives of functions of several variables are functions of the same variables. These functions, in turn, may have partial derivatives, which we will call second partial derivatives (or second-order partial derivatives) of the original function.

So, for example, a function of two variables has four second-order partial derivatives, which are defined and denoted as follows:

A function of three variables has nine second-order partial derivatives:

Similarly, the partial derivatives of the third and higher order of a function of several variables are defined and denoted: the partial derivative of the order of a function of several variables is the first-order partial derivative of the partial derivative of the order of the same function.

For example, the third-order partial derivative of a function is the first-order partial derivative with respect to y of the second-order partial derivative

A second or higher partial derivative taken with respect to several different variables is called a mixed partial derivative.

For example, partial derivatives

are mixed partial derivatives of a function of two variables.

Example. Find second order mixed partial derivatives of a function

Solution. Finding partial derivatives of the first order

Then we find the mixed partial derivatives of the second order

We see that mixed partial derivatives and differing only in the order of differentiation, i.e., in the sequence in which differentiation with respect to various variables is performed, turned out to be identically equal. This result is not accidental. Regarding mixed partial derivatives, the following theorem holds, which we accept without proof.

We continue the favorite topic of mathematical analysis - derivatives. In this article, we will learn how to find partial derivatives of a function of three variables: first derivatives and second derivatives. What do you need to know and be able to master the material? Do not believe it, but, firstly, you need to be able to find the "ordinary" derivatives of a function of one variable - at a high or at least an average level. If it’s really tight with them, then start with a lesson How to find the derivative? Secondly, it is very important to read the article and comprehend and solve, if not all, then most of the examples. If this has already been done, then walk with me with a confident gait, it will be interesting, you will even get pleasure!

Methods and principles of finding partial derivatives of a function of three variables are actually very similar to partial derivative functions of two variables. The function of two variables, I remind you, has the form , where "x" and "y" are independent variables. Geometrically, a function of two variables is a certain surface in our three-dimensional space.

The function of three variables has the form , while the variables are called independentvariables or arguments, the variable is called dependent variable or function. For example: - a function of three variables

And now a little about science fiction films and aliens. You often hear about 4D, 5D, 10D, etc. spaces. Nonsense or not?
After all, the function of three variables implies the fact that all things take place in a four-dimensional space (indeed, there are four variables). The graph of a function of three variables is the so-called hypersurface. It is impossible to imagine it, since we live in a three-dimensional space (length/width/height). So that you are not bored with me, I offer a quiz. I will ask a few questions, and those who wish can try to answer them:

- Is there a fourth, fifth, etc. in the world? measurements in the sense of the philistine understanding of space (length/width/height)?

- Is it possible to build a four-dimensional, five-dimensional, etc. space in the broad sense of the word? That is, to give an example of such a space in our life.

Is it possible to travel to the past?

Is it possible to travel to the future?

- Do aliens exist?

For any question, you can choose one of four answers:
Yes / No (science forbids this) / Science does not forbid / Don't know

Whoever answers all the questions correctly, he most likely possesses some thing ;-)

I will gradually give out answers to questions during the lesson, do not skip the examples!

Actually, they flew. And now the good news: for a function of three variables, the rules of differentiation and the table of derivatives are valid. That is why you need to be good at managing the "ordinary" derivatives of functions one variable. There are very few differences!

Example 1

Solution: It is easy to guess that for a function of three variables there are three partial derivatives of the first order, which are denoted as follows:

Or - partial derivative of "x";
or - partial derivative with respect to "y";
or - partial derivative with respect to "z".

The notation with a stroke is more in use, but the compilers of collections, manuals in the conditions of tasks are very fond of using just cumbersome notations - so don't get lost! Perhaps not everyone knows how to read these "terrible fractions" aloud correctly. Example: should be read as follows: “de u po de x”.

Let's start with the x-derivative: . When we find the partial derivative with respect to , then the variables and are considered constants (constant numbers). And the derivative of any constant, oh, grace, is equal to zero:

Immediately pay attention to the subscript - no one forbids you to mark that they are constants. It’s even more convenient, I recommend that beginners use just such a record, there is less risk of confusion.

(1) We use the properties of the linearity of the derivative, in particular, we take out all the constants from the sign of the derivative. Please note that in the second term, the constant does not need to be taken out: since the “y” is a constant, then it is also a constant. In the term, the "usual" constant 8 and the constant "zet" are taken out of the sign of the derivative.

(2) We find the simplest derivatives, not forgetting that are constants. Next, comb the answer.

Partial derivative . When we find the partial derivative with respect to "y", then the variables and are considered constants:

(1) We use the properties of linearity. And again, note that the terms are constants, which means that nothing needs to be taken out for the sign of the derivative.

(2) We find derivatives, not forgetting that constants. Let's simplify the answer.

And finally, the partial derivative. When we find the partial derivative with respect to "z", then the variables and are considered constants:

General rule obvious and unpretentious: When we find the partial derivativefor any independent variable, thentwo others independent variables are considered constants.

When designing these tasks, you should be extremely careful, in particular, can't lose subscripts(which indicate on which variable differentiation is made). The loss of the index will be a GREAT FAULT. Hmmm…. it's funny if, after such an intimidation, I myself will miss them somewhere)

Example 2

Find partial derivatives of the first order of a function of three variables

This is a do-it-yourself example. Full solution and answer at the end of the lesson.

The two examples considered are quite simple and, having solved several similar problems, even a teapot will adapt to cracking down on them verbally.

To unload, let's return to the first question of the quiz: Is there a fourth, fifth, etc. in the world? measurements in the sense of the philistine understanding of space (length/width/height)?

Correct answer: Science doesn't forbid it.. All fundamental mathematical axiomatics, theorems, mathematical apparatus are beautiful and consistent work in space of any dimension. It is possible that somewhere in the Universe there are hypersurfaces that are not subject to our mind, for example, a four-dimensional hypersurface, which is given by a function of three variables. Or maybe there are hypersurfaces next to us or even we are right in them, just our vision, other sense organs, consciousness are capable of perceiving and comprehending only three dimensions.

Let's get back to the examples. Yes, if someone is heavily loaded with a quiz, it is better to read the answers to the following questions after you learn how to find partial derivatives of a function of three variables, otherwise I will take out the whole brain to you in the course of the article =)

In addition to the simplest Examples 1,2, in practice there are tasks that can be called a small puzzle. Such examples, to my annoyance, fell out of sight when I created the lesson. Partial derivatives of functions of two variables. Making up for lost time:

Example 3

Solution: It seems to be “everything is simple”, but the first impression is deceptive. When finding partial derivatives, many will guess on the coffee grounds and make mistakes.

Let's analyze the example consistently, clearly and clearly.

Let's start with the partial derivative with respect to x. When we find the partial derivative with respect to "x", then the variables are considered constants. Therefore, the index of our function is also a constant. For dummies, I recommend the following solution: on the draft, change the constant to a specific positive integer, for example, to “five”. The result is a function of one variable:
or you can also write it like this:

it power function with complex base (sine). By :

Now remember that , thus:

On a clean copy, of course, the solution should be drawn up like this:

We find the partial derivative with respect to "y", they are considered constants. If "x" is a constant, then it is also a constant. On the draft, we do the same trick: we replace, for example, with 3, "Z" - we will replace it with the same "five". The result is again a function of one variable:

it demonstration function with a complex exponent. By the rule of differentiation of a complex function:

Now remember our replacement:

In this way:

On a clean copy, of course, the design should look nice:

And the mirror case with a partial derivative with respect to "z" (- constants):

With some experience, the analysis can be carried out mentally.

We carry out the second part of the task - we compose a differential of the first order. It is very simple, by analogy with a function of two variables, the first-order differential is written by the formula:

In this case:

And business then. I note that in practical problems, the full differential of the 1st order of a function of three variables is required to be compiled much less frequently than for a function of two variables.

A fun example for a do-it-yourself solution:

Example 4

Find first-order partial derivatives of a function of three variables and make a first-order total differential

Full solution and answer at the end of the lesson. If you have any difficulties, use the considered "chainikov's" algorithm, it is guaranteed to help. And another useful tip - do not hurry. Such examples are not quickly solved even by me.

We digress and analyze the second question: Is it possible to build a four-dimensional, five-dimensional, etc. space in the broad sense of the word? That is, to give an example of such a space in our life.

Correct answer: Yes. And, it's very easy. For example, we add a fourth dimension to the length/width/height - time. Popular four-dimensional space-time and the well-known theory of relativity carefully stolen by Einstein from Lobachevsky, Poincaré, Lorentz and Minkowski. Not everyone knows either. Why did Einstein get the Nobel Prize? There was a terrible scandal in the scientific world, and the Nobel Committee formulated the merit of the plagiarist as follows: "For the general contribution to the development of physics." So that's it. Einstein's C grade brand is pure promotion and PR.

It is easy to add a fifth dimension to the considered four-dimensional space, for example: atmospheric pressure. And so on, so on, so on, as many dimensions you set in your model - there will be so many. In the broad sense of the word, we live in a multidimensional space.

Let's look at a couple more typical tasks:

Example 5

Find first order partial derivatives at a point

Solution: A task in this formulation is often encountered in practice and involves the following two actions:
– you need to find partial derivatives of the first order;
– you need to calculate the values ​​of partial derivatives of the 1st order at the point .

We decide:

(1) We have a complex function, and the first step is to take the derivative of the arc tangent. In doing so, we, in fact, calmly use the tabular formula for the derivative of the arc tangent. By the rule of differentiation of a complex function the result must be multiplied by the derivative of the inner function (embedding): .

(2) We use the properties of linearity.

(3) And we take the remaining derivatives, not forgetting that they are constants.

According to the assignment condition, it is necessary to find the value of the found partial derivative at the point . Substitute the coordinates of the point in the found derivative:

The advantage of this task is the fact that other partial derivatives are found in a very similar way:

As you can see, the solution template is almost the same.

Let's calculate the value of the found partial derivative at the point :

And finally, the derivative with respect to "z":

Ready. The solution could also be formulated in another way: first, find all three partial derivatives, and then calculate their values ​​at the point . But, it seems to me, the above method is more convenient - they just found the partial derivative, and immediately, without leaving the cash register, calculated its value at a point.

It is interesting to note that, geometrically, a point is a very real point in our three-dimensional space. The values ​​of the function, derivatives are already the fourth dimension, and no one knows where it is geometrically located. As they say, no one crawled around the Universe with a tape measure, did not check.

As soon as the philosophical theme has gone again, let's consider the third question: Is it possible to travel into the past?

Correct answer: Not. Traveling into the past contradicts the second law of thermodynamics about the irreversibility of physical processes (entropy). So please do not dive into a pool without water, the event can only be played back in the video =) Folk wisdom has come up with the opposite worldly law for a reason: “Measure seven times, cut once”. Although, in fact, a sad thing, time is unidirectional and irreversible, none of us will look younger tomorrow. And various science fiction films like "Terminator" from a scientific point of view are complete nonsense. It is also absurd from the point of view of philosophy - when the Consequence, returning to the past, can destroy its own Cause. .

More interesting with the derivative with respect to "z", although, it's still almost the same:

(1) We take the constants out of the sign of the derivative.

(2) Here again the product of two functions, each of which depends from the "live" variable "z". In principle, you can use the formula for the derivative of a quotient, but it’s easier to go the other way - to find the derivative of the product.

(3) A derivative is a tabular derivative. The second term contains the already familiar derivative of a complex function.

Example 9

Find partial derivatives of the first order of a function of three variables

This is a do-it-yourself example. Think about how it is more rational to find one or another partial derivative. Full solution and answer at the end of the lesson.

Before proceeding to the final examples of the lesson and consider second order partial derivatives functions of three variables, I will once again cheer everyone up with the fourth question:

Is it possible to travel to the future?

Correct answer: Science doesn't forbid it.. Paradoxically, there is no mathematical, physical, chemical or other natural science law that would prohibit travel to the future! Seems like nonsense? But almost everyone in life had a premonition (and not supported by any logical arguments) that this or that event would happen. And it happened! Where did the information come from? From the future? Thus, fantastic films about traveling into the future, and, by the way, the predictions of all kinds of fortune-tellers, psychics cannot be called such nonsense. At least, science has not refuted this. Everything is possible! So, when I was in school, CDs and flat-panel monitors from movies seemed like incredible fantasy to me.

The well-known comedy "Ivan Vasilyevich Changes His Profession" is half fiction (as a maximum). No scientific law forbade Ivan the Terrible to be in the future, but it is impossible for two peppers to be in the past and perform the duties of a king.

The concept of a function of many variables

Let there be n-variables and each x 1, x 2 ... x n from a certain set x is assigned a definition. the number Z, then on the set x the function Z \u003d f (x 1, x 2 ... x n) of many variables is given.

X - area of ​​\u200b\u200bdefined functions

x 1, x 2 ... x n - independent variable (arguments)

Z - function Example: Z \u003d P x 2 1 * x 2 (Cylinder volume)

Consider Z \u003d f (x; y) - f-tion of 2 variables x (x 1, x 2 replaced by x, y). The results are by analogy transferred to other functions of many variables. The area of ​​\u200b\u200bdefining the function of 2 variables is the entire cord of the square (ooh) or part of it. Mn-in the value of the th function of 2 variables - the surface in a 3-dimensional space.

Techniques for constructing graphs: - Rassm-t section over the surface of the square || coordinate squares.

Example: x \u003d x 0, zn. square X || 0yz y \u003d y 0 0xz Type of function: Z \u003d f (x 0, y); Z=f(x, y 0)

For example: Z=x 2 +y 2 -2y

Z= x 2 +(y-1) 2 -1 x=0 Z=(y-1) 2 -1 y=1 Z= x 2 -1 Z=0 x 2 +(y-1) 2 -1

Parabola circle(center(0;1)

Limits and continuity of functions of two variables

Let Z = f (x; y) be given, then A is the limit of the f-tion in m. (x 0, y 0), if for any arbitrarily small put. number E>0 noun-t positive number b>0, that for all x,y satisfying |x-x 0 |<б; |y-y 0 |<б выполняется нерав-во |f(x,y)-A|

Z \u003d f (x; y) is continuous in t. (x 0, y 0), if: - it is defined in this t .; - has a finite limit at x, tending to x 0 and y to y 0; - this limit = value

functions in t. (x 0, y 0), i.e. limf (x; y) \u003d f (x 0, y 0)

If the function is continuous in each. t. mn-va X, then it is continuous in this area

Differential function, its geomeaning. The use of dif-la in approximate values.

dy=f’(x)∆x – differential function

dy=dx, i.e. dy=f '(x)dx if y=x

From a geologist's point of view, a function differential is an increment in the ordinate of the tangent drawn to the graph of the function at a point with the abscissa x 0

Dif-l is used in the calculation of approx. function values ​​according to the formula: f(x 0 +∆x)~f(x 0)+f’(x 0)∆x

The closer ∆x is to x, the more accurate the result.

Partial derivatives of the first and second order

First order derivative (which is called private)

A. Let x, y be the increments of independent variables x and y at some point from the region X. Then the value equal to z = f(x + x, y + y) = f(x, y) is called the total increment at the point x 0, y 0. If the variable x is fixed, and the variable y is incremented by y, then we get zу = f(x, y, + y) – f(x, y)

The partial derivative of the variable y is defined similarly, i.e.

The partial derivative of a function of 2 variables is found according to the same rules as for functions of one variable.

The difference is that when differentiating a function with respect to the variable x, y is considered const, and when differentiating with respect to y, x is considered const.

Isolated consts are connected to the function with addition/subtraction operations.

The associated consts are connected to the function with multiplication/division operations.

Derivative of isolated const = 0

1.4.Total differential of a function of 2 variables and its applications

Let z = f(x,y), then

tz = - is called a full increment

Partial derivative of the 2nd order

For continuous functions of 2 variables, the mixed partial derivatives of the 2nd order and coincide.

The use of partial derivatives to determine the partial derivatives of max and min functions are called extrema.

A. Points are called max or min z = f(x,y) if there are some segments such that for all x and y from this neighborhood f(x,y)

T. If an extremum point of a function of 2 variables is given, then the value of partial derivatives at this point is equal to 0, i.e. ,

The points at which first-order partial derivatives are called stationary or critical.

Therefore, to find the extremum points of a function of 2 variables, sufficient extremum conditions are used.

Let the function z = f(x,y) be twice differentiable, and let the stationary point,

1) , and maxA<0, minA>0.

1.4.(*)full differential. The geometric meaning of the differential. Application of the differential in approximate calculations

O. Let the function y = f(x) be defined in some neighborhood at the points . A function f(x) is called differentiable at a point if its increment at this point , where is represented in the form (1)

Where A is a constant value independent of , at a fixed point x, - infinitely small at . A relatively linear function A is called the differential of the function f(x) at a point and is denoted by df() or dy.

Thus, expression (1) can be written as ().

The function differential in expression (1) has the form dy = A . Like any linear function, it is defined for any value while the increment of the function must be considered only for those for which + belongs to the domain of the function f(x).

For convenience of notation of the differential, the increment is denoted by dx and is called the differential of the independent variable x. Therefore, the differential is written as dy = Adx.

If the function f(x) is differentiable at every point of some interval, then its differential is a function of two variables - the point x and the variable dx:

T. In order for the function y = g(x) to be differentiable at some point , it is necessary and sufficient that it has a derivative at this point, while

(*)Proof. Need.

Let the function f(x) be differentiable at the point , i.e., . Then

Therefore, the derivative f'() exists and is equal to A. Hence dy = f'()dx

Adequacy.

Let there be a derivative f'(), i.e. = f'(). Then the curve y = f(x) is a tangent segment. To calculate the value of a function at a point x, take a point in some of its neighborhood, such that it is not difficult to find f() and f’()/

Partial derivatives of functions of two variables.
Concept and examples of solutions

In this lesson, we will continue our acquaintance with the function of two variables and consider, perhaps, the most common thematic task - finding partial derivatives of the first and second order, as well as the total differential of the function. Part-time students, as a rule, face partial derivatives in the 1st year in the 2nd semester. Moreover, according to my observations, the task of finding partial derivatives is almost always found in the exam.

In order to effectively study the following material, you necessary be able to more or less confidently find the "usual" derivatives of a function of one variable. You can learn how to handle derivatives correctly in the lessons How to find the derivative? and Derivative of a complex function. We also need a table of derivatives of elementary functions and differentiation rules, it is most convenient if it is at hand in printed form. You can find reference material on the page Mathematical formulas and tables.

Let's quickly repeat the concept of a function of two variables, I will try to limit myself to the bare minimum. A function of two variables is usually written as , with the variables being called independent variables or arguments.

Example: - a function of two variables.

Sometimes the notation is used. There are also tasks where the letter is used instead of a letter.

From a geometric point of view, a function of two variables is most often a surface of three-dimensional space (a plane, a cylinder, a ball, a paraboloid, a hyperboloid, etc.). But, in fact, this is already more analytical geometry, and we have mathematical analysis on the agenda, which my university teacher never let me write off is my “horse”.

We turn to the question of finding partial derivatives of the first and second orders. I have some good news for those of you who have had a few cups of coffee and are in the mood for unimaginably difficult material: partial derivatives are almost the same as the "ordinary" derivatives of a function of one variable.

For partial derivatives, all the rules of differentiation and the table of derivatives of elementary functions are valid. There are only a couple of small differences that we will get to know right now:

... yes, by the way, for this topic I did create small pdf book, which will allow you to "fill your hand" in just a couple of hours. But, using the site, you, of course, will also get the result - just maybe a little slower:

Example 1

Find partial derivatives of the first and second order of a function

First, we find the partial derivatives of the first order. There are two of them.

Notation:
or - partial derivative with respect to "x"
or - partial derivative with respect to "y"

Let's start with . When we find the partial derivative with respect to "x", then the variable is considered a constant (constant number).

Comments on the actions taken:

(1) The first thing we do when finding the partial derivative is to conclude all function in parentheses under the dash with subscript.

Attention important! Subscripts DO NOT LOSE in the course of the solution. In this case, if you draw a “stroke” somewhere without, then the teacher, at least, can put it next to the task (immediately bite off part of the score for inattention).

(2) Use the rules of differentiation , . For a simple example like this one, both rules can be applied in the same step. Pay attention to the first term: since is considered a constant, and any constant can be taken out of the sign of the derivative, then we take it out of brackets. That is, in this situation, it is no better than a regular number. Now let's look at the third term: here, on the contrary, there is nothing to take out. Since it is a constant, it is also a constant, and in this sense it is no better than the last term - the “seven”.

(3) We use tabular derivatives and .

(4) We simplify, or, as I like to say, "combine" the answer.

Now . When we find the partial derivative with respect to "y", then the variableconsidered a constant (constant number).

(1) We use the same differentiation rules , . In the first term we take out the constant beyond the sign of the derivative, in the second term nothing can be taken out because it is already a constant.

(2) We use the table of derivatives of elementary functions. Mentally change in the table all "X" to "Y". That is, this table is equally valid for (and indeed for almost any letter). In particular, the formulas we use look like this: and .

What is the meaning of partial derivatives?

At their core, 1st order partial derivatives resemble "ordinary" derivative:

- this is functions, which characterize rate of change functions in the direction of the axes and respectively. So, for example, the function characterizes the steepness of "climbs" and "slopes" surfaces in the direction of the abscissa axis, and the function tells us about the "relief" of the same surface in the direction of the ordinate axis.

! Note : here refers to directions that are parallel coordinate axes.

For the sake of better understanding, let's consider a specific point of the plane and calculate the value of the function (“height”) in it:
- and now imagine that you are here (ON THE VERY surface).

We calculate the partial derivative with respect to "x" at a given point:

The negative sign of the "X" derivative tells us about descending functions at a point in the direction of the x-axis. In other words, if we make a small-small (infinitesimal) step towards the tip of the axis (parallel to this axis), then go down the slope of the surface.

Now we find out the nature of the "terrain" in the direction of the y-axis:

The derivative with respect to "y" is positive, therefore, at a point along the axis, the function increases. If it’s quite simple, then here we are waiting for an uphill climb.

In addition, the partial derivative at a point characterizes rate of change functions in the relevant direction. The greater the resulting value modulo- the steeper the surface, and vice versa, the closer it is to zero, the flatter the surface. So, in our example, the "slope" in the direction of the abscissa axis is steeper than the "mountain" in the direction of the ordinate axis.

But those were two private paths. It is quite clear that from the point at which we are, (and in general from any point of the given surface) we can move in some other direction. Thus, there is an interest in compiling a general "navigation chart" that would tell us about the "landscape" of the surface. if possible at every point scope of this function in all available ways. I will talk about this and other interesting things in one of the next lessons, but for now let's get back to the technical side of the issue.

We systematize the elementary applied rules:

1) When we differentiate by , then the variable is considered a constant.

2) When differentiation is carried out according to, then is considered a constant.

3) The rules and the table of derivatives of elementary functions are valid and applicable for any variable (or any other) with respect to which differentiation is carried out.

Step two. We find partial derivatives of the second order. There are four of them.

Notation:
or - the second derivative with respect to "x"
or - the second derivative with respect to "y"
or - mixed derivative "x by y"
or - mixed derivative "Y with X"

There are no problems with the second derivative. In simple terms, the second derivative is the derivative of the first derivative.

For convenience, I will rewrite the first-order partial derivatives already found:

First we find the mixed derivatives:

As you can see, everything is simple: we take the partial derivative and differentiate it again, but in this case, already by “y”.

Similarly:

In practical examples, you can focus on the following equality:

Thus, through mixed derivatives of the second order, it is very convenient to check whether we have found the partial derivatives of the first order correctly.

We find the second derivative with respect to "x".
No inventions, we take and differentiate it by "X" again:

Similarly:

It should be noted that when finding , you need to show increased attention, since there are no miraculous equalities to test them.

The second derivatives also find wide practical application, in particular, they are used in the problem of finding extrema of a function of two variables. But everything has its time:

Example 2

Calculate the first order partial derivatives of the function at the point . Find derivatives of the second order.

This is an example for self-solving (answers at the end of the lesson). If you have difficulty differentiating roots, go back to the lesson How to find the derivative? In general, pretty soon you will learn how to find similar derivatives on the fly.

We fill our hand with more complex examples:

Example 3

Check that . Write the total differential of the first order.

Solution: We find partial derivatives of the first order:

Pay attention to the subscript: next to the "x" it is not forbidden to write in brackets that it is a constant. This mark can be very useful for beginners to make it easier to navigate the solution.

Further comments:

(1) We take out all the constants outside the sign of the derivative. In this case, and , and, hence, their product is considered a constant number.

(2) Do not forget how to properly differentiate the roots.

(1) We take all the constants out of the sign of the derivative, in this case the constant is .

(2) Under the prime, we have the product of two functions, therefore, we need to use the product differentiation rule .

(3) Do not forget that is a complex function (although the simplest of the complex ones). We use the corresponding rule: .

Now we find mixed derivatives of the second order:

This means that all calculations are correct.

Let's write the total differential. In the context of the task under consideration, it makes no sense to tell what the total differential of a function of two variables is. It is important that this very differential very often needs to be written down in practical problems.

Total First Order Differential functions of two variables has the form:

In this case:

That is, in the formula you just need to stupidly just substitute the already found partial derivatives of the first order. Differential icons and in this and similar situations, if possible, it is better to write in numerators:

And at the repeated request of readers, full differential of the second order.

It looks like this:

CAREFULLY find the "single-letter" derivatives of the 2nd order:

and write down the "monster", carefully "attaching" the squares, the product and not forgetting to double the mixed derivative:

It's okay if something seemed difficult, you can always return to derivatives later, after you pick up the differentiation technique:

Example 4

Find first order partial derivatives of a function . Check that . Write the total differential of the first order.

Consider a series of examples with complex functions:

Example 5

Find partial derivatives of the first order of the function .

Solution:

Example 6

Find first order partial derivatives of a function .
Write down the total differential.

This is an example for self-solving (answer at the end of the lesson). I won't post the complete solution because it's quite simple.

Quite often, all of the above rules are applied in combination.

Example 7

Find first order partial derivatives of a function .

(1) We use the rule of differentiating the sum

(2) The first term in this case is considered a constant, since there is nothing in the expression that depends on "x" - only "y". You know, it's always nice when a fraction can be turned into zero). For the second term, we apply the product differentiation rule. By the way, in this sense, nothing would change if a function were given instead - it is important that here the product of two functions, EACH of which depends on "X", and therefore, you need to use the rule of differentiation of the product. For the third term, we apply the rule of differentiation of a complex function.

(1) In the first term, both the numerator and the denominator contain a “y”, therefore, you need to use the rule for differentiating the quotient: . The second term depends ONLY on "x", which means it is considered a constant and turns into zero. For the third term, we use the rule of differentiation of a complex function.

For those readers who courageously made it almost to the end of the lesson, I’ll tell you an old Mekhmatov anecdote for detente:

Once an evil derivative appeared in the space of functions and how it went to differentiate everyone. All functions scatter in all directions, no one wants to turn! And only one function does not escape anywhere. The derivative approaches it and asks:

"Why aren't you running away from me?"

- Ha. But I don't care, because I'm "e to the power of x", and you can't do anything to me!

To which the evil derivative with an insidious smile replies:

- This is where you are wrong, I will differentiate you by “y”, so be zero for you.

Who understood the joke, he mastered the derivatives, at least for the "troika").

Example 8

Find first order partial derivatives of a function .

This is a do-it-yourself example. A complete solution and a sample design of the problem are at the end of the lesson.

Well, that's almost all. Finally, I cannot help but please mathematicians with one more example. It's not even about amateurs, everyone has a different level of mathematical training - there are people (and not so rare) who like to compete with more difficult tasks. Although, the last example in this lesson is not so much complicated as cumbersome in terms of calculations.

The general principle of finding second-order partial derivatives of a function of three variables is similar to the principle of finding second-order partial derivatives of a function of two variables.

In order to find the partial derivatives of the second order, you must first find the partial derivatives of the first order or, in another notation:

There are nine partial derivatives of the second order.

The first group is the second derivatives with respect to the same variables:

Or - the second derivative with respect to "x";

Or - the second derivative with respect to "y";

Or - the second derivative with respect to "z".

The second group is mixed partial derivatives of the 2nd order, there are six of them:

Or - mixed derivative "by x y";

Or - mixed derivative "by y x";

Or - mixed derivative "by x z";

Or - mixed derivative "po zet x";

Or - mixed derivative "by game z";

Or - mixed derivative "po z y".

As in the case of a function of two variables, when solving problems, one can focus on the following equalities of mixed second-order derivatives:

Note: Strictly speaking, this is not always the case. For the equality of mixed derivatives, it is necessary to fulfill the requirement of their continuity.

Just in case, a few examples of how to read this disgrace out loud:

- "two strokes twice a y";

- “de two y po de zet square”;

- “two strokes on x on z”;

- “de two y po de z po de y”.

Example 10

Find all first and second order partial derivatives for a function of three variables:

.

Solution: First, we find the partial derivatives of the first order:

We take the found derivative

and differentiate it by "y":

We take the found derivative

and differentiate it by "x":

Equality is done. Good.

We deal with the second pair of mixed derivatives.

We take the found derivative

and differentiate it by "z":

We take the found derivative

and differentiate it by "x":

Equality is done. Good.

Similarly, we deal with the third pair of mixed derivatives:

Equality is done. Good.

After the work done, it can be guaranteed that, firstly, we correctly found all partial derivatives of the 1st order, and secondly, we also correctly found the mixed partial derivatives of the 2nd order.

It remains to find three more partial derivatives of the second order, here, in order to avoid errors, you should concentrate as much as possible:

Ready. Again, the task is not so much difficult as voluminous. The solution can be shortened and referred to as equalities of mixed partial derivatives, but in this case there will be no verification. So it's better to take the time and find all derivatives (besides, this may be required by the teacher), or, in extreme cases, check on a draft.

Example 11

Find all first and second order partial derivatives for a function of three variables

.

This is a do-it-yourself example.

Solutions and answers:

Example 2:Solution:

Example 4:Solution: Let us find partial derivatives of the first order.

We compose the total differential of the first order:

Example 6:Solution: M(1, -1, 0):

Example 7:Solution: Let us calculate the partial derivatives of the first order at the pointM(1, 1, 1):

Example 9:Solution:

Example 11:Solution: Let's find partial derivatives of the first order:

Let's find partial derivatives of the second order:

.

Integrals

8.1. Indefinite integral. Detailed Solution Examples

Let's start studying the topic Indefinite integral", and also analyze in detail examples of solutions to the simplest (and not quite) integrals. As usual, we will limit ourselves to the minimum theory that is in numerous textbooks, our task is to learn how to solve integrals.

What do you need to know to successfully master the material? In order to cope with integral calculus, you need to be able to find derivatives, at least at an average level. It will not be superfluous experience if you have several dozen, or better, a hundred independently found derivatives behind you. At the very least, you should not be confused by the task of differentiating the simplest and most common functions.

It would seem, where are the derivatives at all, if we are talking about integrals in the article?! And here's the thing. The fact is that finding derivatives and finding indefinite integrals (differentiation and integration) are two mutually inverse actions, such as addition / subtraction or multiplication / division. Thus, without a skill and some kind of experience in finding derivatives, unfortunately, one cannot advance further.

In this regard, we will need the following methodological materials: Derivative table and Table of integrals.

What is the difficulty of studying indefinite integrals? If in derivatives there are strictly 5 rules of differentiation, a table of derivatives and a fairly clear algorithm of actions, then in integrals everything is different. There are dozens of integration methods and techniques. And, if the integration method was initially chosen incorrectly (that is, you don’t know how to solve it), then the integral can literally be “pricked” for literally days, like a real rebus, trying to notice various tricks and tricks. Some even like it.

By the way, we quite often heard from students (not humanities) an opinion like: “I never had an interest in solving the limit or derivative, but integrals are a completely different matter, it’s exciting, there is always a desire to “crack” a complex integral” . Stop. Enough black humor, let's move on to these very indefinite integrals.

Since there are many ways to solve, then where does a teapot start studying indefinite integrals? In integral calculus, in our opinion, there are three pillars or a kind of "axis" around which everything else revolves. First of all, you should have a good understanding of the simplest integrals (this article).

Then you need to work out the lesson in detail. THIS IS THE MOST IMPORTANT RECEPTION! Perhaps even the most important article of all articles devoted to integrals. And thirdly, be sure to read integration by parts, because it integrates a wide class of functions. If you master at least these three lessons, then there are already “not two”. You can be forgiven for not knowing integrals of trigonometric functions, integrals of fractions, integrals of fractional rational functions, integrals of irrational functions (roots), but if you “get into a puddle” on the replacement method or the integration by parts method, then it will be very, very bad.

So, let's start simple. Let's look at the table of integrals. As in derivatives, we notice several integration rules and a table of integrals of some elementary functions. Any tabular integral (and indeed any indefinite integral) has the form:

Let's get straight to the notation and terms:

- integral icon.

- integrand function (written with the letter "s").

– differential icon. What it is, we will consider very soon. The main thing is that when writing the integral and during the solution, it is important not to lose this icon. There will be a noticeable flaw.

is the integrand or "stuffing" of the integral.

– antiderivative function.

– . There is no need to be heavily loaded with terms, the most important thing here is that in any indefinite integral, a constant is added to the answer.

To solve an indefinite integral means to findset of antiderivative functions from the given integrand

Let's take a look at the entry again:

Let's look at the table of integrals.

What's happening? Our left parts are turning to other functions: .

Let's simplify our definition:

Solve the indefinite integral - it means to TURN it into an indefinite (up to a constant) function , using some rules, techniques and a table.

Take, for example, the table integral . What happened? The symbolic record has turned into a set of antiderivative functions.

As in the case of derivatives, in order to learn how to find integrals, it is not necessary to be aware of what an integral is, or an antiderivative function from a theoretical point of view. It is enough just to carry out transformations according to some formal rules. So, in case it is not at all necessary to understand why the integral turns into exactly. You can take this and other formulas for granted. Everyone uses electricity, but few people think about how electrons run along the wires.

Since differentiation and integration are opposite operations, then for any antiderivative that is found correctly, the following is true:

In other words, if the correct answer is differentiated, then the original integrand must be obtained.

Let's go back to the same table integral .

Let's verify the validity of this formula. We take the derivative of the right side:

is the original integrand.

By the way, it became clearer why a constant is always assigned to a function. When differentiating, a constant always turns into zero.

Solve the indefinite integral it means to find lots of all antiderivatives, and not some single function. In the considered tabular example, , , , etc. - all these functions are the solution of the integral . There are infinitely many solutions, so they write briefly:

Thus, any indefinite integral is easy enough to check. This is some compensation for a large number of integrals of different types.

Let's move on to specific examples. Let's start, as in the study of the derivative, with two rules of integration:

- constant C can (and should) be taken out of the integral sign.

– the integral of the sum (difference) of two functions is equal to the sum (difference) of two integrals. This rule is valid for any number of terms.

As you can see, the rules are basically the same as for derivatives. Sometimes they are called linearity properties integral.

Example 1

Find the indefinite integral.

Run a check.

Solution: It is more convenient to convert it like.

(1) Applying the rule . Don't forget to write down the differential icon dx under each integral. Why under each? dxis a full multiplier. If you paint in detail, then the first step should be written as follows:

.

(2) According to the rule we take all the constants out of the signs of the integrals. Note that in the last term tg 5 is a constant, we also take it out.

In addition, at this step we prepare the roots and degrees for integration. In the same way as in differentiation, the roots must be represented in the form . Roots and degrees that are located in the denominator - move up.

Note: unlike derivatives, roots in integrals do not always need to be reduced to the form , and move the degrees up.

For example, - this is a ready-made tabular integral, which has already been calculated before you, and all sorts of Chinese tricks like completely unnecessary. Similarly: - this is also a tabular integral, there is no point in representing a fraction in the form . Study the table carefully!

(3) All integrals are tabular. We carry out the transformation using the table, using the formulas: , and

for a power function - .

It should be noted that the table integral is a special case of the formula for a power function: .

Constant C just add it once at the end of the expression

(rather than putting them after each integral).

(4) We write the result obtained in a more compact form, when all degrees of the form

again represent as roots, and the powers with a negative exponent are reset back to the denominator.

Examination. In order to perform the check, you need to differentiate the received answer:

Initial integrand, i.e., the integral was found correctly. From what they danced, to that they returned. It's good when the story with the integral ends just like that.

From time to time there is a slightly different approach to checking the indefinite integral, when not the derivative, but the differential is taken from the answer:

.

As a result, we obtain not an integrand, but an integrand.

Do not be afraid of the concept of differential.

The differential is the derivative multiplied by dx.

However, it is not theoretical subtleties that are important to us, but what to do next with this differential. The differential is revealed as follows: icon d remove, put a stroke on the right above the bracket, assign a multiplier at the end of the expression dx :

Received original integrand, that is, the integral is found correctly.

As you can see, the differential comes down to finding the derivative. I like the second way of checking less, since I have to additionally draw large brackets and drag the differential icon dx until the end of the test. Although it is more correct, or "more solid", or something.

In fact, it was possible to keep silent about the second method of verification. The point is not in the method, but in the fact that we have learned to open the differential. Again.

The differential is revealed as follows:

1) icon d remove;

2) put a stroke on the right above the bracket (the designation of the derivative);

3) at the end of the expression we assign a factor dx .

For example:

Remember this. We will need the considered technique very soon.

Example 2

.

When we find an indefinite integral, we ALWAYS try to check Moreover, there is a great opportunity for this. Not all types of problems in higher mathematics are a gift from this point of view. It does not matter that verification is often not required in control tasks, no one, and nothing prevents it from being carried out on a draft. An exception can be made only when there is not enough time (for example, at the test, exam). Personally, I always check integrals, and I consider the lack of verification to be a hack and a poorly completed task.

Example 3

Find the indefinite integral:

. Run a check.

Solution: Analyzing the integral, we see that under the integral we have the product of two functions, and even the exponentiation of the whole expression. Unfortunately, in the field of integral battle No good and comfortable formulas for integrating the product and the quotient as: or .

Therefore, when a product or a quotient is given, it always makes sense to see if it is possible to transform the integrand into a sum? The considered example is the case when it is possible.

First, we give the complete solution, the comments will be below.

(1) We use the good old formula for the square of the sum for any real numbers, getting rid of the degree above the common bracket. outside the brackets and applying the abbreviated multiplication formula in the opposite direction: .

Example 4

Find the indefinite integral

Run a check.

This is an example for self-solving. Answer and complete solution at the end of the lesson.

Example 5

Find the indefinite integral

. Run a check.

In this example, the integrand is a fraction. When we see a fraction in the integrand, the first thought should be the question: “Is it possible to somehow get rid of this fraction, or at least simplify it?”.

We notice that the denominator contains a lone root of "x". One in the field is not a warrior, which means that you can divide the numerator into the denominator term by term:

We do not comment on actions with fractional powers, since they have been repeatedly discussed in articles on the derivative of a function.

If you are still confused by such an example as

and no one gets the right answer,

Also note that the solution skips one step, namely applying the rules , . Usually, with a certain experience in solving integrals, these rules are considered an obvious fact and are not described in detail.

Example 6

Find the indefinite integral. Run a check.

This is an example for self-solving. Answer and complete solution at the end of the lesson.

In the general case, with fractions in integrals, everything is not so simple, additional material on the integration of fractions of some types can be found in the article: Integration of some fractions. But, before moving on to the above article, you need to read the lesson: Replacement method in indefinite integral. The fact is that summing a function under a differential or a variable change method is key point in the study of the topic, since it occurs not only "in pure assignments for the replacement method", but also in many other varieties of integrals.

Solutions and answers:

Example 2: Solution:

Example 4: Solution:

In this example, we used the reduced multiplication formula

Example 6: Solution:

The method of changing a variable in an indefinite integral. Solution examples

In this lesson, we will get acquainted with one of the most important and most common tricks that is used in the course of solving indefinite integrals - the change of variable method. For successful mastering of the material, initial knowledge and integration skills are required. If there is a feeling of an empty full teapot in integral calculus, then you should first familiarize yourself with the material Indefinite integral. Solution examples, where it is explained in an accessible form what an integral is and basic examples for beginners are analyzed in detail.

Technically, the method of changing a variable in an indefinite integral is implemented in two ways:

– Bringing the function under the sign of the differential.

– The actual change of variable.

In fact, it's the same thing, but the design of the solution looks different. Let's start with a simpler case.