Concept of a function of many variables

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

X – area of ​​function definition

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

Z – function Example: Z=P x 2 1 *x 2 (Cylinder volume)

Consider Z=f(x;y) – the function of 2 variables (x 1, x 2 replaced by x,y). The results are transferred by analogy to other functions of many variables. The area for determining the function of 2 variables is the entire cord (oh) or part of it. The number of values ​​of the function of 2 variables is a surface in 3-dimensional space.

Techniques for constructing graphs: - Consider the cross-section of the surface in squares || coordinate squares.

Example: x = x 0, zn. square X || 0уz y = y 0 0хz Type of function: Z=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 surround(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 function in t.(x 0 ,y 0), if for any arbitrarily small set. number E>0 is a positive number b>0, which for all x, y satisfying |x-x 0 |<б; |y-y 0 |<б выполняется нерав-во |f(x,y)-A|

Z=f(x;y) is continuous in a t. (x 0 ,y 0) if: - it is defined in this t.; - has a final 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)=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 geom meaning. Application of differential in approximate values.

dy=f’(x)∆x – differential function

dy=dx, i.e. dy=f ’(x)dx if y=x

From a geological point of view, the differential of a function is the increment of the ordinate of the tangent drawn to the graph of the function at the point with the abscissa x 0

Dif-l is used in calculating approx. values ​​of the function 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 partial)

A. Let x, y be the increments of the 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 we fix the variable x and give the increment y to the variable y, then we get zу = f(x,y,+ y) – f(x,y)

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

The partial derivative of a function of 2 variables is found using 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, it is considered const.

Isolated const are connected to a function using addition/subtraction operations.

Bound const are connected to a function by multiplication/division operations.

Derivative of isolated const = 0

1.4.Complete differential function of 2 variables and its applications

Let z = f(x,y), then

tz = - called full increment

2nd order partial derivative

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

The application of partial derivatives to the determination of 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 the 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 a stationary point,

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

1.4.(*)Full differential. Geometric meaning of differential. Application of differential in approximate calculations

A. Let the function y = f(x) be defined in a certain neighborhood at the points. A function f(x) is said to be differentiable at a point if its increment at this point is , where it is presented in the form (1)

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

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

The differential of the function 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 definition of the function f(x).

For convenience of writing 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 each point of a certain 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 have a derivative at this point, and

(*)Proof. Necessity.

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 neighborhood of it, such that it is not difficult to find f() and f’()/

Partial derivatives of a function 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, 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.

To effectively study the material below, 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 will 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 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.

From a geometric point of view, a function of two variables most often represents a surface in three-dimensional space (plane, cylinder, sphere, paraboloid, hyperboloid, etc.). But, in fact, this is more analytical geometry, and on our agenda is mathematical analysis, which my university teacher never let me write off and 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 a simple example like this, 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. I will talk about this and other interesting things in one of the following 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. 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, 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 you to solve on your own (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 better at 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 take all constants outside 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 very 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 the “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). I won't give you a 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 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 joke for relief:

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.

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

(Where y= const),

(Where x= const).

Therefore, partial derivatives are calculated using formulas and rules for calculating derivatives of functions of one variable, while considering the other variable constant.

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

If it’s hard to concentrate to keep track of where the constant is in the function, then in the draft solution of the example, instead of a variable with a fixed value, you can substitute any number - then you can quickly calculate the partial derivative as the ordinary derivative of a function of one variable. You just need to remember to return the constant (a variable with a fixed value) to its place when finishing the final design.

The property of partial derivatives described above follows from the definition of a partial derivative, which may appear in exam questions. Therefore, to familiarize yourself with the definition below, you can open the theoretical reference.

Concept of continuity of 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

Difference (2) is called the total increment of the function z(it is obtained as a result of increments of both arguments).

Let the function be given z= f(x, y) and period

If the function change z occurs when only one of the arguments changes, for example, x, with a fixed value of another argument y, then the function will receive an increment

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

Considering a function change z depending on changing only one of the arguments, we effectively change 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 indicated by one of the symbols

(4)

The partial increment is determined 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 is fixed: in this case it is simply a certain number that is a factor (as in the case of the ordinary derivative) of the variable with which we find the partial derivative. If the fixed variable is not multiplied by the variable with which we find the partial derivative, then this lonely constant, no matter to what extent, as in the case of the 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 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 fixed x the derivative of the first term is found as the derivative of the exponential function, and the second - as the derivative of a constant:

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

You can check the solution to partial derivative problems at online partial derivative calculator .

Example 3. Find partial derivatives of a function

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 function sign);

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

You can check the solution to partial derivative problems at online partial derivative calculator .

The partial derivatives of a function 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 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 determined and calculated under the assumption that only one of the independent variables changes, while the others are fixed.

Example 4. Find partial derivatives of a function

.

Solution. y And z fixed:

x And z fixed:

x And y fixed:

Find partial derivatives yourself and then look at the 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 is the same as the derivative of a function of one variable, is the rate of change of the function relative to a change in one of the arguments.

Example 8. Quantitative value of flow P railway passengers can be expressed by the function

Where P– number of passengers, N– number of residents of correspondent points, R– distance between points.

Partial derivative of a function P By R, equal

shows that the decrease in passenger flow is inversely proportional to the square of the distance between corresponding points with the same number of residents in points.

Partial derivative P By N, equal

shows that the increase in passenger flow is proportional to twice the number of residents of settlements at the same distance between points.

You can check the solution to partial derivative problems at online partial derivative calculator .

Full differential

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

The sum of partial differentials for 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 complete differential of a function

Solution. The result of using formula (7):

A function that has a total differential at every point of a certain domain is said to be differentiable in that domain.

Find the total differential yourself and then look at the solution

Just as in the case of a function of one variable, the differentiability of a function in a certain domain implies its continuity in this domain, 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 the 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 at and .

Higher order partial derivatives

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

Partial derivatives are used in problems involving 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:

First order partial derivatives

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

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

Second order partial derivatives

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

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

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 complex function

a) Let $ z (t) = f(x(t), y(t)) $, then the derivative of a 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 implicit 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 $$ z"_x = - \frac(F"_x)(F"_z); z"_y = - \frac(F"_y)( F"_z) $$

Examples of solutions

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 consider $ y $ to be 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 $y$, we define $y$ by 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 detailed solution. You will be able to view the progress of the calculation and gain information. This will help you get your grade from your teacher in a timely manner!

Answer

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

Example 2

Find the partial derivatives of the 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 $ to be a constant value: $$ z"_y = (e^(xy))"_y = e^(xy) \cdot (xy)"_y = xe^(xy) $$ Knowing the first derivatives, we similarly find the second. Set $y$ to a constant: $$ z""_(xx) = (z"_x)"_x = (ye^(xy))"_x = (y)"_x e^(xy) + y(e^(xy))"_x = 0 + ye^(xy)\cdot (xy)"_x = y^2e^(xy) $$ We set $ x $ to a 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 all that remains is to find the mixed derivative. You can differentiate $ z"_x $ by $ y $, and you can differentiate $ z"_y $ by $ 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 the implicit function $ F(x,y,z) = 0 $. Find first order partial derivatives.

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's summarize how finding partial derivatives differs from finding “ordinary” derivatives of a function of one variable:

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

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

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

Step two. We find second-order partial derivatives. 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 "by x igrek"

Or - mixed derivative "by igrek 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 already found first-order partial derivatives:

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:

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

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 wonderful equalities to check.

Example 2

Find the first and second order partial derivatives of the function

This is an example for you to solve on your own (answer at the end of the lesson).

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

Let's move on to 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 take all constants outside 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 very differential very often needs to be written down in practical problems.

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

In this case:

That is, you just need to 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:

Example 4

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

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.

Let's look at a series of examples that involve complex functions.

Example 5

Find the first order partial derivatives of the function.

(1) We apply the rule of differentiation of complex functions . From class Derivative of a complex function a very important point should be remembered: when we turn a sine (external function) into a cosine using the table, then we have an embedding (internal function) does not change.

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

Likewise:

Let's write down the first order complete differential:

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). I won't give you a 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 differentiation of 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, nothing would have changed in the algorithm if a function had been 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.