Partial derivatives are used in assignments with functions of several variables. The rules for finding are exactly the same as for functions of one variable, with the only difference being that one of the variables must be considered a constant (constant number) at the time of differentiation.

Formula

Partial derivatives for a function of two variables $ z(x,y) $ are written in the following form $ z"_x, z"_y $ and are found using the formulas:

Partial derivatives of the first order

$$ z"_x = \frac(\partial z)(\partial x) $$

$$ z"_y = \frac(\partial z)(\partial y) $$

Partial derivatives of the second order

$$ z""_(xx) = \frac(\partial^2 z)(\partial x \partial x) $$

$$ z""_(yy) = \frac(\partial^2 z)(\partial y \partial y) $$

mixed derivative

$$ z""_(xy) = \frac(\partial^2 z)(\partial x \partial y) $$

$$ z""_(yx) = \frac(\partial^2 z)(\partial y \partial x) $$

Partial derivative of a compound function

a) Let $ z (t) = f(x(t), y(t)) $, then the derivative of the complex function is determined by the formula:

$$ \frac(dz)(dt) = \frac(\partial z)(\partial x) \cdot \frac(dx)(dt) + \frac(\partial z)(\partial y) \cdot \frac (dy)(dt) $$

b) Let $ z (u,v) = z(x(u,v),y(u,v)) $, then the partial derivatives of the function are found by the formula:

$$ \frac(\partial z)(\partial u) = \frac(\partial z)(\partial x) \cdot \frac(\partial x)(\partial u) + \frac(\partial z)( \partial y) \cdot \frac(\partial y)(\partial u) $$

$$ \frac(\partial z)(\partial v) = \frac(\partial z)(\partial x) \cdot \frac(\partial x)(\partial v) + \frac(\partial z)( \partial y) \cdot \frac(\partial y)(\partial v) $$

Partial derivatives of an implicitly given function

a) Let $ F(x,y(x)) = 0 $, then $$ \frac(dy)(dx) = -\frac(f"_x)(f"_y) $$

b) Let $ F(x,y,z)=0 $, then $$ z"_x = - \frac(F"_x)(F"_z); z"_y = - \frac(F"_y)( F"_z) $$

Solution examples

Example 1

Find first order partial derivatives $ z (x,y) = x^2 - y^2 + 4xy + 10 $

Solution

To find the partial derivative with respect to $ x $, we will assume that $ y $ is a constant value (number): $$ z"_x = (x^2-y^2+4xy+10)"_x = 2x - 0 + 4y + 0 = 2x+4y $$ To find the partial derivative of a function with respect to $ y $, define $ y $ as a constant: $$ z"_y = (x^2-y^2+4xy+10)"_y = -2y+4x $$ If you cannot solve your problem, then send it to us. We will provide a detailed solution. You will be able to familiarize yourself with the progress of the calculation and gather information. This will help you get a credit from the teacher in a timely manner!

Answer

$$ z"_x = 2x+4y; z"_y = -2y+4x $$

Example 2

Find partial derivatives of a second order function $ z = e^(xy) $

Solution

First you need to find the first derivatives, and then knowing them, you can find the second order derivatives. Let $ y $ be a constant: $$ z"_x = (e^(xy))"_x = e^(xy) \cdot (xy)"_x = ye^(xy) $$ Let us now set $ x $ as a constant value: $$ z"_y = (e^(xy))"_y = e^(xy) \cdot (xy)"_y = xe^(xy) $$ Knowing the first derivatives, we similarly find the second ones. Set $y$ constant: $$ z""_(xx) = (z"_x)"_x = (ye^(xy))"_x = (y)"_x e^(xy) + y(e^(xy))"_x = 0 + ye^(xy)\cdot (xy)"_x = y^2e^(xy) $$ Set $ ​​x $ constant: $$ z""_(yy) = (z"_y)"_y = (xe^(xy))"_y = (x)"_y e^(xy) + x(e^(xy))"_y = 0 + x^2e^(xy) = x^2e^(xy) $$ Now it remains to find the mixed derivative. You can differentiate $ z"_x $ with respect to $ y $, or you can differentiate $ z"_y $ with respect to $ x $, since by the theorem $ z""_(xy) = z""_(yx) $ $$ z""_(xy) = (z"_x)"_y = (ye^(xy))"_y = (y)"_y e^(xy) + y (e^(xy))"_y = ye^(xy)\cdot (xy)"_y = yxe^(xy) $$

Answer

$$ z"_x = ye^(xy); z"_y = xe^(xy); z""_(xy) = yxe^(xy) $$

Example 4

Let $ 3x^3z - 2z^2 + 3yz^2-4x+z-5 = 0 $ define an implicit function $ F(x,y,z) = 0 $. Find partial derivatives of the first order.

Solution

We write the function in the format: $ F(x,y,z) = 3x^3z - 2z^2 + 3yz^2-4x+z-5 = 0 $ and find the derivatives: $$ z"_x (y,z - const) = (x^3 z - 2z^2 + 3yz^2-4x+z-5)"_x = 3 x^2 z - 4 $$ $$ z"_y (x,y - const) = (x^3 z - 2z^2 + 3yz^2-4x+z-5)"_y = 3z^2 $$

Answer

$$ z"_x = 3x^2 z - 4; z"_y = 3z^2; $$

Let a function be given. Since x and y are independent variables, one of them can change while the other remains unchanged. Let's increment the independent variable x while keeping the value of y unchanged. Then z will receive an increment, which is called the partial increment of z by x and is denoted by . So, .

Similarly, we obtain a partial increment of z with respect to y: .

The total increment of the function z is determined by the equality .

If there is a limit, then it is called the partial derivative of the function at the point with respect to the variable x and is denoted by one of the symbols:

.

Partial derivatives with respect to x at a point are usually denoted by the symbols .

The partial derivative of with respect to the variable y is defined and denoted in a similar way:

Thus, the partial derivative of a function of several (two, three or more) variables is defined as the derivative of a function of one of these variables, subject to the constancy of the values ​​of the remaining independent variables. Therefore, the partial derivatives of a function are found according to the formulas and rules for calculating the derivatives of a function of one variable (in this case, x or y, respectively, are considered a constant value).

Partial derivatives are also called partial derivatives of the first order. They can be considered as functions of . These functions can have partial derivatives, which are called second-order partial derivatives. They are defined and denoted as follows:

; ;

; .

Differentials of 1st and 2nd order of a function of two variables.

The total differential of a function (formula 2.5) is called the first-order differential.

The formula for calculating the total differential is as follows:

(2.5) or , Where ,

partial differentials of the function .

Let the function have continuous partial derivatives of the second order. The second order differential is determined by the formula . Let's find it:

From here: . Symbolically it is written like this:

.

UNDEFINITE INTEGRAL.

Antiderivative of a function, indefinite integral, properties.

The function F(x) is called primitive for a given function f(x), if F"(x)=f(x), or, which is the same, if dF(x)=f(x)dx.

Theorem. If a function f(x), defined in some interval (X) of finite or infinite length, has one antiderivative, F(x), then it also has infinitely many antiderivatives; they are all contained in the expression F(x)+C, where C is an arbitrary constant.

The set of all antiderivatives for a given function f(x), defined in some interval or on some segment of finite or infinite length, is called indefinite integral from the function f(x) [or from the expression f(x)dx ] and is denoted by the symbol .

If F(x) is one of the antiderivatives for f(x), then by the antiderivative theorem

, where C is an arbitrary constant.

By the definition of the antiderivative F "(x)=f(x) and, therefore, dF(x)=f(x) dx. In the formula (7.1), f(x) is called the integrand, and f(x) dx is called the integrand expression.

To summarize, what is the difference between finding partial derivatives and finding “ordinary” derivatives of a function of one variable:

1) When we find the partial derivative, That variable considered a constant.

2) When we find the partial derivative, That variable considered a constant.

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

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

Designations:

Or - the second derivative with respect to "x"

Or - the second derivative with respect to "y"

Or - mixed derivative "with respect to x y"

Or - mixed derivative "with respect to x"

There is nothing complicated about the concept of the second derivative. In simple terms, The second derivative is the derivative of the first derivative.

For clarity, 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:

For practical examples, when all partial derivatives are continuous, the following equality holds:

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 check.

Example 2

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

This is an example for self-solving (answer at the end of the lesson).

With some experience, partial derivatives from examples No. 1, 2 will be solved by you orally.

Let's move on to 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.

The total first-order differential of a function of two variables has the form:

In this case:

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

Example 4

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

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

Consider a series of examples that include complex functions.

Example 5

Find partial derivatives of the first order of the function .

(1) We apply the rule of differentiation of a complex function . From the lesson Derivative of a compound function a very important point should be remembered: when we turn the sine (external function) into a cosine according to the table, then the investment (internal function) we have does not change.

(2) Here we use the property of the roots: , take the constant out of the sign of the derivative, and represent the root in the form necessary for differentiation.

Similarly:

We write the total differential of the first order:

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 you can turn a fraction into zero.)

For the second term, we apply the product differentiation rule. By the way, nothing would change in the algorithm if a function were given instead - it is important that here we have product of two functions, EACH of which depends on "x", so you need to use the product differentiation rule. For the third term, we apply the rule of differentiation of a complex function.

Let a function of two variables be given. Let's increment the argument and leave the argument unchanged. Then the function will receive an increment, which is called a partial increment with respect to the variable and is denoted:

Similarly, by fixing the argument and giving the argument an increment, we get a partial increment of the function with respect to the variable:

The value is called the full increment of the function at the point.

Definition 4. The partial derivative of a function of two variables with respect to one of these variables is the limit of the ratio of the corresponding partial increment of the function to the increment of the given variable when the latter tends to zero (if this limit exists). The partial derivative is denoted as: or, or.

Thus, by definition, we have:

Partial derivatives of a function are calculated according to the same rules and formulas as a function of one variable, taking into account that when differentiating with respect to a variable, it is considered constant, and when differentiating with respect to a variable, it is considered constant.

Example 3. Find partial derivatives of functions:

Solution. a) To find we assume a constant value and differentiate as a function of one variable:

Similarly, assuming a constant value, we find:

Definition 5. The total differential of a function is the sum of the products of the partial derivatives of this function and the increments of the corresponding independent variables, i.e.

Given that the differentials of the independent variables coincide with their increments, i.e. , the formula for the total differential can be written as

Example 4. Find the total differential of a function.

Solution. Since, then by the formula of the total differential we find

Partial derivatives of higher orders

Partial derivatives are also called partial derivatives of the first order or first partial derivatives.

Definition 6. Second-order partial derivatives of a function are partial derivatives of first-order partial derivatives.

There are four second-order partial derivatives. They are designated as follows:

The partial derivatives of the 3rd, 4th and higher orders are defined similarly. For example, for a function we have:

Partial derivatives of the second or higher order taken with respect to different variables are called mixed partial derivatives. For a function, these are derivatives. Note that in the case when mixed derivatives are continuous, then equality takes place.

Example 5. Find second order partial derivatives of a function

Solution. First order partial derivatives for this function are found in example 3:

Differentiating and with respect to the variables x and y, we get

Each partial derivative (over x and by y) of a function of two variables is the ordinary derivative of a function of one variable with a fixed value of the other variable:

(Where y= const),

(Where x= const).

Therefore, partial derivatives are calculated from formulas and rules for calculating derivatives of functions of one variable, while considering the other variable as a constant (constant).

If you do not need an analysis of examples and the minimum theory necessary for this, but you only need a solution to your problem, then proceed to online partial derivative calculator .

If it’s hard to focus on keeping track of where the constant is in the function, then you can substitute any number in the draft solution of the example instead of a variable with a fixed value - then you can quickly calculate the partial derivative as the ordinary derivative of a function of one variable. It is only necessary not to forget to return the constant (a variable with a fixed value) to its place when finishing.

The property of partial derivatives described above follows from the definition of a partial derivative, which can be found in exam questions. Therefore, to get acquainted with the definition below, you can open the theoretical reference.

The concept of continuity of a function z= f(x, y) at a point is defined similarly to this concept for a function of one variable.

Function z = f(x, y) is called continuous at a point if

The difference (2) is called the total increment of the function z(it is obtained by incrementing both arguments).

Let the function z= f(x, y) and dot

If the function change z occurs when only one of the arguments changes, for example, x, with a fixed value of the other argument y, then the function will be incremented

called partial increment of the function f(x, y) By x.

Considering the function change z depending on the change of only one of the arguments, we actually pass to a function of one variable.

If there is a finite limit

then it is called the partial derivative of the function f(x, y) by argument x and is denoted by one of the symbols

(4)

The partial increment is defined similarly z By y:

and partial derivative f(x, y) By y:

(6)

Example 1

Solution. We find the partial derivative with respect to the variable "x":

(y fixed);

We find the partial derivative with respect to the variable "y":

(x fixed).

As you can see, it does not matter to what extent the variable that is fixed: in this case, it is just some number that is a factor (as in the case of the usual derivative) with the variable by which we find the partial derivative. If the fixed variable is not multiplied by the variable with respect to which we find the partial derivative, then this lonely constant, no matter to what extent, as in the case of an ordinary derivative, vanishes.

Example 2 Given a function

Find Partial Derivatives

(by x) and (by y) and calculate their values ​​at the point A (1; 2).

Solution. At a fixed y the derivative of the first term is found as the derivative of the power function ( table of derivative functions of one variable):

.

At a fixed x the derivative of the first term is found as the derivative of the exponential function, and the second - as the derivative of the constant:

Now we calculate the values ​​of these partial derivatives at the point A (1; 2):

You can check the solution of problems with partial derivatives on online partial derivative calculator .

Example 3 Find Partial Derivatives of Functions

Solution. In one step we find

(y x, as if the argument of sine were 5 x: in the same way, 5 appears before the sign of the function);

(x is fixed and is in this case a factor at y).

You can check the solution of problems with partial derivatives on online partial derivative calculator .

The partial derivatives of functions of three or more variables are defined similarly.

If each set of values ​​( x; y; ...; t) independent variables from the set D corresponds to one specific value u from many E, That u is called a function of variables x, y, ..., t and denote u= f(x, y, ..., t).

For functions of three or more variables, there is no geometric interpretation.

Partial derivatives of a function of several variables are also defined and calculated under the assumption that only one of the independent variables changes, while the others are fixed.

Example 4 Find Partial Derivatives of Functions

.

Solution. y And z fixed:

x And z fixed:

x And y fixed:

Find partial derivatives on your own and then see solutions

Example 5

Example 6 Find partial derivatives of a function.

The partial derivative of a function of several variables has the same mechanical meaning as the derivative of a function of one variable, is the rate at which the function changes relative to a change in one of the arguments.

Example 8 flow quantity P railway passengers can be expressed as a function

Where P- the number of passengers, N- the number of residents of the corresponding points, R– distance between points.

Partial derivative of a function P By R equal to

shows that the decrease in the flow of passengers is inversely proportional to the square of the distance between the corresponding points for the same number of inhabitants in the points.

Partial derivative P By N equal to

shows that the increase in the flow of passengers is proportional to twice the number of inhabitants of settlements with the same distance between points.

You can check the solution of problems with partial derivatives on online partial derivative calculator .

Full differential

The product of the partial derivative and the increment of the corresponding independent variable is called the partial differential. Partial differentials are denoted as follows:

The sum of partial differentials over all independent variables gives the total differential. For a function of two independent variables, the total differential is expressed by the equality

(7)

Example 9 Find the full differential of a function

Solution. The result of using formula (7):

A function that has a total differential at every point of some domain is called differentiable in that domain.

Find the total differential on your own and then see the solution

Just as in the case of a function of one variable, the differentiability of a function in a certain region implies its continuity in this region, but not vice versa.

Let us formulate without proof a sufficient condition for the differentiability of a function.

Theorem. If the function z= f(x, y) has continuous partial derivatives

in a given region, then it is differentiable in this region and its differential is expressed by formula (7).

It can be shown that, just as in the case of a function of one variable, the differential of the function is the main linear part of the increment of the function, so in the case of a function of several variables, the total differential is the main, linear with respect to increments of independent variables, part of the total increment of the function.

For a function of two variables, the total increment of the function has the form

(8)

where α and β are infinitesimal for and .

Partial derivatives of higher orders

Partial derivatives and functions f(x, y) are themselves some functions of the same variables and, in turn, may have derivatives with respect to different variables, which are called partial derivatives of higher orders.