Total differential of a function of two variables at a point. Total differential of a function of several variables

As you can see, to find the differential you need to multiply the derivative by dx. This allows you to immediately write down the corresponding table for differentials from the table of formulas for derivatives.

Total differential for a function of two variables:

The total differential for a function of three variables is equal to the sum of partial differentials: d f(x,y,z)=d x f(x,y,z)dx+d y f(x,y,z)dy+d z f(x,y,z)dz

Definition . A function y=f(x) is called differentiable at a point x 0 if its increment at this point can be represented as ∆y=A∆x + α(∆x)∆x, where A is a constant and α(∆x) – infinitesimal as ∆x → 0.
The requirement that a function be differentiable at a point is equivalent to the existence of a derivative at this point, and A=f’(x 0).

Let f(x) be differentiable at the point x 0 and f "(x 0)≠0, then ∆y=f'(x 0)∆x + α∆x, where α= α(∆x) →0 at ∆x →0. The quantity ∆y and each term on the right side are infinitesimal quantities for ∆x→0. , that is, α(∆x)∆x is an infinitesimal more high order, than f’(x 0)∆x.
, that is, ∆y~f’(x 0)∆x. Consequently, f’(x 0)∆x represents the main and at the same time linear relative to ∆x part of the increment ∆y (linear - meaning containing ∆x to the first power). This term is called the differential of the function y=f(x) at the point x 0 and is denoted dy(x 0) or df(x 0). So, for arbitrary values of x
dy=f′(x)∆x. (1)
Set dx=∆x, then
dy=f′(x)dx. (2)

Example. Find derivatives and differentials of these functions.
a) y=4 tan2 x
Solution:

differential:
b)
Solution:

differential:
c) y=arcsin 2 (lnx)
Solution:

differential:
G)
Solution:
=
differential:

Example. For the function y=x 3 find an expression for ∆y and dy for some values of x and ∆x.
Solution. ∆y = (x+∆x) 3 – x 3 = x 3 + 3x 2 ∆x +3x∆x 2 + ∆x 3 – x 3 = 3x 2 ∆x+3x∆x 2 +∆x 3 ; dy=3x 2 ∆x (we took the main linear part ∆y relative to ∆x). In this case, α(∆x)∆x = 3x∆x 2 + ∆x 3.

Partial derivatives of a function of two variables.
Concept and examples of solutions

On 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, encounter partial derivatives in the 1st year in the 2nd semester. Moreover, according to my observations, the task of finding partial derivatives almost always appears on the exam.

For effective learning the following material for you necessary be able to more or less confidently find “ordinary” derivatives of functions of one variable. You can learn how to handle derivatives correctly in lessons How to find the derivative? And Derivative of a complex function. We also need a table of derivatives elementary functions and the rules of differentiation, it is most convenient if it is at hand in printed form. You can get 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: – function of two variables.

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

WITH geometric point In terms of vision, a function of two variables most often represents a surface of three-dimensional space (plane, cylinder, sphere, paraboloid, hyperboloid, etc.). But, in fact, this is more analytical geometry, and on our agenda mathematical analysis, who never let me cheat, my university teacher is my strong point.

Let's move on to the question of finding partial derivatives of the first and second orders. I have some good news for those who have had a few cups of coffee and are tuning in to some incredibly difficult material: partial derivatives are almost the same as “ordinary” derivatives of a function of one variable.

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

...yes, by the way, for this topic I created small pdf book, which will allow you to “get your teeth into” in just a couple of hours. But by using the site, you will certainly get the same result - just maybe a little slower:

Example 1

Find the first and second order partial derivatives of the function

First, let's find the first-order partial derivatives. There are two of them.

Designations:
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”, the variable is considered a constant (constant number).

Comments on the actions performed:

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

Attention, important! WE DO NOT LOSE subscripts during the solution process. In this case, if you draw a “stroke” somewhere without , then the teacher, at a minimum, can put it next to the assignment (immediately bite off part of the point for inattention).

(2) We use the rules of differentiation , . For simple example like this one, both rules can easily be applied in one step. Pay attention to the first term: since is considered a constant, and any constant can be taken out of the derivative sign, then we put it out of brackets. That is, in this situation it is no better than an ordinary 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 - “seven”.

(3) We use tabular derivatives and .

(4) Let’s simplify, or, as I like to say, “tweak” 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 the constant out of the sign of the derivative, in the second term we can’t take anything out since it is already a constant.

(2) We use the table of derivatives of elementary functions. Let’s mentally change all the “X’s” in the table to “I’s”. 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?

In essence, 1st order partial derivatives resemble "ordinary" derivative:

- This functions, which characterize rate of change functions in the direction of the and axes, respectively. So, for example, the function characterizes the steepness of “rises” 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 we mean directions that parallel coordinate axes.

For the purpose of better understanding, let’s consider a specific point on the plane and calculate the value of the function (“height”) at it:
– and now imagine that you are here (ON THE surface).

Let's calculate the partial derivative with respect to "x" at a given point:

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

Now we find out the nature of the “terrain” in the direction of the ordinate axis:

The derivative with respect to the “y” is positive, therefore, at a point in the direction of the axis the function increases. To put it simply, here we are waiting for an uphill climb.

In addition, the partial derivative at a point characterizes rate of change functions in the corresponding 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 we are at, (and in general from any point on a given surface) we can move in some other direction. Thus, there is an interest in creating a general "navigation map" that would inform us about the "landscape" of the surface if possible at every point domain of definition of this function along all available paths. About this and others interesting things I’ll tell you in one of the next lessons, but for now let’s return to the technical side of the issue.

Let us systematize the elementary applied rules:

1) When we differentiate with respect to , the variable is considered a constant.

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

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

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

Designations:
or – second derivative with respect to “x”
or – second derivative with respect to “Y”
or - mixed derivative of “x by igr”
or - mixed derivative of "Y"

There are no problems with the second derivative. Speaking in simple language, the second derivative is the derivative of the first derivative.

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

First, let's find mixed derivatives:

As you can see, everything is simple: we take the partial derivative and differentiate it again, but in this case - this time according to the “Y”.

Likewise:

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

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

Find the second derivative with respect to “x”.
No inventions, let's take it and differentiate it by “x” again:

Likewise:

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

Second derivatives also find wide practical applications, 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 second order derivatives.

This is an example for independent decision(answers at the end of the lesson). If you have difficulty differentiating roots, return to the lesson How to find the derivative? In general, pretty soon you will learn to find such derivatives “on the fly.”

Let's get our hands on more complex examples:

Example 3

Check that . Write down the first order total differential.

Solution: Find the first order partial derivatives:

Pay attention to the subscript: , next to the “X” it is not forbidden to write in parentheses that it is a constant. This note can be very useful for beginners to make it easier to navigate the solution.

Further comments:

(1) We move all constants beyond the sign of the derivative. In this case, and , and, therefore, their product is considered a constant number.

(2) Don’t forget how to correctly differentiate roots.

(1) We take all 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 left, therefore, we need to use the rule for differentiating the product .

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

Now we find mixed derivatives of the second order:

This means that all calculations were performed correctly.

Let's write down 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 same differential very often needs to be written down in practical problems.

First order total differential function of two variables has the form:

In this case:

That is, you just need to stupidly substitute the already found first-order partial derivatives into the formula. In this and similar situations, it is best to write differential signs in numerators:

And according to repeated requests from readers, second order complete differential.

It looks like this:

Let's CAREFULLY find “one-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 seems difficult; you can always come back to derivatives later, after you've mastered the differentiation technique:

Example 4

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

Let's look at a series of examples with complex functions:

Example 5

Find the first order partial derivatives of the function.

Solution:

Example 6

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

This is an example for you to solve on your own (answer at the end of the lesson). Complete solution I don’t give it 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 for differentiating the sum

(2) The first term in this case is considered a constant, since there is nothing in the expression that depends on the “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 have changed if a function had been given instead - the important thing is that here product of two functions, EACH of which depends on "X", and therefore, you need to use the product differentiation rule. For the third term, we apply the rule of differentiation of a complex function.

(1) The first term in both the numerator and denominator contains a “Y”, therefore, you need to use the rule for differentiating quotients: . The second term depends ONLY on “x”, which means it is considered a constant and turns to zero. For the third term we use the rule for differentiating 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 relaxation:

One day, an evil derivative appeared in the space of functions and started to differentiate everyone. All functions are scattered in all directions, no one wants to transform! And only one function does not run away. The derivative approaches her and asks:

- Why don’t you run away from me?

- Ha. But I don’t care, because I am “e to the power of X”, and you won’t do anything to me!

To which the evil derivative with an insidious smile replies:

- This is where you are mistaken, I will differentiate you by “Y”, so you should be a zero.

Whoever understood the joke has mastered derivatives, at least to the “C” level).

Example 8

Find first order partial derivatives of a function .

This is an example for you to solve on your own. The complete solution and example of the problem are at the end of the lesson.

Well, that's almost all. Finally, I can’t help but please mathematics lovers with one more example. It's not even about amateurs, everyone has a different level of mathematical preparation - 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 complex as it is cumbersome from a computational point of view.

Definition: Full differential function several variables is the sum of all its partial differentials:

Example 1: .

Solution:

Since the partial derivatives of this function are equal to:

Then we can immediately write down the partial differentials of these functions:

, ,

Then the complete differential of the function will look like:

Example 2 Find the complete differential of a function

Solution:

This function is complex, i.e. can be represented as

Finding partial derivatives:

Full differential:

The analytical meaning of the total differential is that the total differential of a function of several variables is main part full increment this function that is, there is an approximate equality: ∆z≈dz.

It is necessary, however, to remember that these approximate equalities are valid only for small differentials dx and dy of the arguments of the function z=f(x,y).

The use of the total differential in approximate calculations is based on the use of the formula ∆z≈dz.

Indeed, if in this formula the increment ∆z of the function is represented in the form and the total differential in the form , then we get:

≈ ,

The resulting formula can be used to approximately find the “new” value of a function of two variables, which it takes for sufficiently small increments of both of its arguments.

Example. Find the approximate value of a function , with the following values of its arguments: 1.01, .

Solution.

Substituting the partial derivatives of the functions found earlier into the formula, we get:

When substituting the values x=1, ∆х=0.01, y=2, ∆у=0.02 we get:

Scalar field.

If at each point of a certain region of space D the function U(p)=U(x,y,z) is specified, then they say that a scalar field is specified in the region D.

If, for example, U(x,y,z) denotes the temperature at the point M(x,y,z), then they say that a scalar temperature field is specified. If region D is filled with liquid or gas and U(x,y,z) denotes pressure, then there is a scalar pressure field. If the location of charges or massive bodies in space is given, then we talk about a potential field.

The scalar field is called stationary, if the function U(x,y,z) does not change over time: U(x,y,z) ≠ f(t).

Any stationary field is characterized by:

1) level surface of the scalar field

2) the rate of change of the field in a given direction.

Level surface scalar field is called locus points at which the function U(x,y,z) takes a constant value, that is, U(x,y,z) = const. The collection of these points forms a certain surface. If we take a different constant, we get a different surface.

Example: Let a scalar field be given. An example of such a field is the field electric potential spot electric charge(+q). Here the level surfaces will be equipotential surfaces , that is, spheres in the center of which there is a charge that creates a field.

The direction of greatest increase in a scalar function is given by a vector called gradient and is indicated by the symbol (or ).

The gradient of a function is found through the partial derivatives of this function and is always perpendicular to the level surface of the scalar field at a given point:

, Where