Private derivatives are applied in tasks with functions of several variables. The rules of location are exactly the same as for the functions of one variable, with the difference only in the fact that one of the variables should be considered at the time of differentiation by constant (constant).

Formula

Private derivatives for the function of two variables $ z (x, y) $ are written in the following form $ z "_x, z" _y $ and are located according to the formulas:

Private derivatives of first order

$$ Z "_X \u003d \\ FRAC (\\ Partial z) (\\ Partial x) $$

$$ Z "_Y \u003d \\ FRAC (\\ Partial z) (\\ Partial y) $$

Second-order private derivatives

$$ z "" _ (xx) \u003d \\ FRAC (\\ Partial ^ 2 z) (\\ Partial x \\ Partial x) $$

$$ z "" _ (yy) \u003d \\ FRAC (\\ Partial ^ 2 z) (\\ Partial Y \\ Partial y) $$

Mixed derivative

$$ z "" _ (XY) \u003d \\ FRAC (\\ Partial ^ 2 z) (\\ Partial x \\ Partial y) $$

$$ Z "" _ (yx) \u003d \\ FRAC (\\ Partial ^ 2 z) (\\ Partial Y \\ Partial X) $$

Partial derivative of complex function

a) Let $ z (t) \u003d f (x (t), y (t)) $, then the derivative of the complex function is determined by the formula:

$$ \\ FRAC (DZ) (DT) \u003d \\ FRAC (\\ Partial Z) (\\ Partial X) \\ CDOT \\ FRAC (DX) (DT) + \\ FRAC (\\ Partial Z) (\\ Partial Y) \\ CDOT \\ FRAC (DY) (DT) $$

b) Let $ z (u, v) \u003d z (x (u, v), y (u, v)) $, then the partial derivatives are in the formula:

$$ \\ FRAC (\\ Partial z) (\\ Partial U) \u003d \\ FRAC (\\ Partial z) (\\ Partial x) \\ CDOT \\ FRAC (\\ Partial x) (\\ Partial U) + \\ FRAC (\\ Partial Z) ( \\ Partial y) \\ CDOT \\ FRAC (\\ Partial Y) (\\ Partial U) $$

$$ \\ FRAC (\\ Partial Z) (\\ Partial V) \u003d \\ FRAC (\\ Partial Z) (\\ Partial X) \\ CDOT \\ FRAC (\\ Partial X) (\\ Partial V) + \\ FRAC (\\ Partial Z) ( \\ Partial Y) \\ Cdot \\ FRAC (\\ Partial Y) (\\ Partial V) $$

Private derivatives implicitly specified function

a) Let $ f (x, y (x)) \u003d 0 $, then $$ \\ FRAC (DY) (DX) \u003d - \\ FRAC (F "_x) (f" _y) $$

b) Let $ f (x, y, z) \u003d 0 $, then $$ z "_x \u003d - \\ FRAC (f" _x) (f "_z); z" _Y \u003d - \\ FRAC (F "_Y) ( F "_z) $$

Examples of solutions

Each private derivative (by x. and in y.) The functions of two variables is an ordinary derivative of the function of one variable with a fixed value of another variable:

(Where y.\u003d const)

(Where x.\u003d const).

Therefore, private derivatives are calculated by formulas and rules for calculating derived functions of one variable , considering the other variable constant (constant).

If you do not need the analysis of the examples and the minimum of the theory necessary for this, and only the solution of your task is necessary, then go to calculator of private derivatives online .

If it is difficult to focus to keep track of where in the function of the constant, then in the draft example of the example instead of a variable with a fixed value to substitute any number - then it will be possible to quickly calculate the private derivative as an ordinary derivative function of one variable. It is only necessary not to forget when it is possible to return to the location of the constant (variable with a fixed value).

The following property of private derivatives described above follows from the definition of a private derivative, which can be caught in examination issues. Therefore, to familiarize yourself with the definition below, you can open theoretical help.

The concept of continuity function z.= f.(x., y.) At the point is determined similarly to this concept for the function of one variable.

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

The difference (2) is called the full increment of the function z.(It is obtained as a result of the increments of both arguments).

Let the function are specified z.= f.(x., y.) and point

If the function is changed z.occurs when changing only one of the arguments, for example, x., with a fixed value of another argument y.then the function will receive increment

called private increment f.(x., y.) by x..

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

If there is a finite limit

then it is called a private derivative function f.(x., y.) By argument x.and is indicated by one of the characters

(4)

Similarly, private increments are determined z.by y.:

and private derivative f.(x., y.) by y.:

(6)

Example 1.

Decision. We find a private derivative on the "X" variable:

(y.fixed);

We find a private derivative in the "Igrek" variable:

(x.fixed).

As can be seen, it does not matter to what extent a variable that is fixed: In this case, it is simply a certain number that is a multiplier (as in the case of a common derivative) with a variable, which we find a private derivative. If a fixed variable is not multiplied by a variable in which we find a private derivative, then this single constant is indifferent to what extent, as in the case of a normal derivative, it turns to zero.

Example 2.Dana feature

Find private derivatives

(by ICSU) and (on Igrek) and calculate their values \u200b\u200bat the point BUT (1; 2).

Decision. With fixed y. The derivative of the first term is as a derivative of the power function ( table of derived functions of one variable):

.

With fixed x. The derivative of the first term is as a derivative of the indicative function, and the second is as a derivative of constant:

Now we calculate the values \u200b\u200bof these private derivatives at the point BUT (1; 2):

Check the solution of tasks with private derivatives calculator private derivatives online .

Example 3. Find private derived functions

Decision. In one step we find

(y. x.as if the argument of sinus was 5 x.: In the same way 5 turns out to be a feature function);

(x. Fixed and is in this case a multiplier when y.).

Check the solution of tasks with private derivatives calculator private derivatives online .

Similarly, private derivatives of three or more variables are determined.

If each set of values \u200b\u200b( x.; y.; ...; t.) independent variables from set D.corresponds to one definite value u.from set E.T. u.called the function of variables x., y., ..., t.and denote u.= f.(x., y., ..., t.).

For the functions of three and more variables, geometric interpretation does not exist.

Private derivatives of several variables are determined and are also calculated as an assumption that only one of the independent variables changes, while others are fixed.

Example 4. Find private derived functions

.

Decision. y. and z. Fixed:

x. and z. Fixed:

x. and y. Fixed:

Find private derivatives independently and then see solutions

Example 5.

Example 6.Find private derived functions.

The private derivative of the function of several variables has the same mechanical meaning as the derivative function of one variable - This is the speed of changing the function relative to the change of one of the arguments.

Example 8. Quantitative value of flow Prail passengers can be expressed by function

where P- the number of passengers, N.- the number of residents of the corresponding items, R.- Distance between items.

Private derivative function Pby R.equal

it shows that the decrease in the flux of passengers is inversely proportional to the square of the distance between the corresponding items at the same number of residents in points.

Private derivative Pby N.equal

it shows that an increase in the flow of passengers in proportion to the doubled number of residents of settlements at the same distance between the points.

Check the solution of tasks with private derivatives calculator private derivatives online .

Full differential

The product of the private derivative on the increment of the corresponding independent variable is called private differential. Private differentials are referred to as:

The amount of private differentials on all independent variables gives a complete differential. For the function of two independent variables, the full differential is expressed by equality

(7)

Example 9.Find a full differential function

Decision. The result of using formula (7):

A function that has a full differential at each point of some area is called differentiable in this area.

Find a full differential yourself, and then see the decision

As in the case of a function of one variable, from differentiability of the function in some region it follows its continuity in this area, but not vice versa.

We formulate without evidence a sufficient condition of the differentiability of the function.

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

in this area, it is differentiated in this area 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 function is the main linear part of the increment of the function, and in the case of the function of several variables, the full differential is the main, linear relative to increments of independent variables part of the full increment of the function.

For the function of two variables, the full increment of the function is

(8)

where α and β are infinitely small with and.

Private derivatives of higher orders

Private derivatives and functions f.(x., y.) They themselves are some functions of the same variables and, in turn, may have derivatives according to different variables, which are called private derivatives of higher orders.

Let the function of two variables be specified. We give an argument increment, and the argument will be left unchanged. Then the function will receive an increment, which is called private increment by variable and is indicated:

Similarly, fixing the argument and giving the argument to the increment, we obtain the private increment of the function by the variable:

The value is called the full prure function at the point.

Definition 4. The private derivative of the function of two variables by one of these variables is the limit of the ratio of the corresponding particular increase of the function to increment to this variable, when the latter tends to zero (if this limit exists). It is indicated by the private derivative so: or, or.

Thus, by definition, we have:

Private derivatives are calculated according to the same rules and formulas as the function of one variable, it is taken into account that during differentiation according to a variable, it is considered a constant, and when differentiating on a variable constant is considered.

Example 3. Find private derived functions:

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

Similarly, considering the permanent values, we find:

Definition 5. The total differential function is called the amount of works of private derivatives of this function on the increments of the corresponding independent variables, i.e.

Considering that differentials of independent variables coincide with their increments, i.e. , full differential formula can be written as

Example 4. Find a full differential function.

Decision. Since, according to the full differential formula, we find

Private derivatives of higher orders

Private derivatives and are called private first-order derivatives or first private derivatives.

Determination 6. Partial derivatives of the function are called private derivatives from private derivatives of first-order.

Second-order private derivatives four. They are indicated as follows:

Similarly, the private derivatives of the 3rd, 4th and higher orders are determined. For example, for the function we have:

Private derivatives of a second or higher order, taken according to various variables, are called mixed private derivatives. For functions are derivatives. Note that in the case when mixed derivatives are continuous, there is equality.

Example 5. Find partial derivatives of the second order function

Decision. Private derivatives of the first order for this function found in Example 3:

Differentiating and via variables and y, we get

Let the function specify. Since X and Y are independent variables, then one of them may vary, and the other to save its value. We give an independent variable x increment, keeping the value of y unchanged. Then z will receive an increment called the private increment of Z according to X and is indicated. So, .

Similarly, we obtain a private increment Z on Y :.

The complete increment of the function Z is determined by the equality.

If there is a limit, it is called a private derivative function at the point in the variable x and is indicated by one of the characters:

.

Private derivatives by x at the point are usually denoted by symbols .

Similarly, it is determined and denoted by the private derivative of the variable y:

Thus, the particular derivative of the function of several (two, three and more) variables is defined as a derivative of one of these variables, provided that the values \u200b\u200bof the remaining independent variables are constant. Therefore, private derivatives are in the formulas and rules for calculating the derivatives of one variable (at the same time, respectively, x or y are considered a constant value).

Private derivatives and are called private first-order derivatives. They can be considered as functions from. These functions may have private derivatives, which are called second-order private derivatives. They are determined and referred to as follows:

; ;

; .

Differentials 1 and 2 orders of function of two variables.

The full differential function (formula 2.5) is called the first order differential.

The formula for calculating the complete differential is as follows:

(2.5) or where

private differentials function.

Suppose that the function has continuous private second-order derivatives. The second order differential is determined by the formula. Find him:

From here: . It is symbolically written in this way:

.

Uncertain integral.

Pred-like function, an indefinite integral, properties.

Function F (x) is called predo-shapedfor this function f (x), if f "(x) \u003d f (x), or, that is the same if Df (x) \u003d f (x) dx.

Theorem. If the function f (x), determined in some interval (x) of a finite or infinite length, has one primitive, f (x), then it has and infinitely many primitive; All of them are contained in the expression F (x) + C, where C is an arbitrary constant.

The combination of all the primary functions f (x) defined in some interval or on some segment of the final or infinite length is called uncertain integral From the function f (x) [or from the expression F (x) dx] and is indicated by the symbol.

If f (x) is one of the primitive for F (X), then according to the primary theorem

where with there is an arbitrary constant.

By definition of the primitive f "(x) \u003d f (x) and, therefore, df (x) \u003d f (x) dx. In formula (7.1), F (x) is called the guideline function, and F (x) DX is aimed expression.

Let us summarize what is the finding of private derivatives from finding "ordinary" derivatives of one variable:

1) When we find a private derivative T. variable it is considered a constant.

2) When we find a private derivative T. variable it is considered a constant.

3) Rules and table of derivative elementary functions are valid and applicable to any variable ( , or some other), on which differentiation is conducted.

Step second. We find private derivatives of the second order. Four them.

Designations:

Or - the second derivative of "X"

Or is the second derivative of "Igarek"

Or - mixedderivative "by ixes"

Or - mixedthe derivative "on the Iks"

In the concept of the second derivative there is nothing complicated. In simple language, the second derivative is a derivative of the first derivative.

For clarity, I will rewrite the first-order private derivatives already found:

We will find mixed derivatives first:

As you can see, everything is simple: we take a private derivative and differentiate it again, but in this case it is already "Igarek".

Similarly:

For practical examples, when all private derivatives are continuous, the following equality is true:

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

We find the second derivative of the "X".

No inventions take And differentiate it by "X" again:

Similarly:

It should be noted that when you find, you need to show increased attentionSince no wonderful equivals for checking exist.

Example 2.

Find private derivatives of the first and second order function

This is an example for an independent solution (answer at the end of the lesson).

With a certain experience, private derivatives from examples number 1,2 will be solved orally.

Go to more complex examples.

Example 3.

Check that. Write a complete first order differential.

Solution: Find private derivatives of the first order:

Pay attention to the substitution index :, Next to the "Iks" is not rebored in brackets, which is a constant. This mark can be very useful for beginners to make it easier to navigate in the decision.

Further comments:

(1) We endure all the constants for the sign of the derivative. In this case, and, it means, their work is considered a constant number.

(2) Do not forget how to differentiate roots.

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

(2) Under the stroke, we left the product of two functions, therefore, you need to use the derivation of the work .

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

Now we find mixed derivatives of the second order:

So, all calculations are fulfilled.

We write a full differential. In the context of the considered task, it makes no sense to tell that such a complete differential function of two variables. It is important that this most differential is very often required to record in practical tasks.

The full differential of the first order of the function of two variables is:

.

In this case:

That is, in the formula you just need to substitute the first-order private derivatives already found. Differentially icons and in this and similar situations can be better recorded in numerators if possible.

Example 4.

Find private derivatives of first-order function . Check that. Write a complete first order differential.

This is an example for an independent solution. Complete solution and sample design task - at the end of the lesson.

Consider a series of examples that include complex functions.

Example 5.

(1) Apply the differentiation rule of a complex function . From lesson Derivative complex functionit should be remembered a very important point: when we turn the sinus on the table (external function) into the cosine, then the investment (internal function) does not change.

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

Similarly:

We write the full differential of the first order:

Example 6.

Find private derivatives of first-order function .

Write a full differential.

This is an example for an independent solution (answer at the end of the lesson). Complete solution do not bring, as it is quite simple

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

Example 7.

Find partial derivatives of the first order of the function.

(1) Use the amount differentiation rule.

(2) The first term in this case is considered to be a constant, since there is nothing dependent on "Iks" in the expression - only "ignorax".

(You know, always nice when the fraction can turn into zero).

For the second component, we use a derivation of the product differentiation. By the way, in the algorithm it would not have changed, if instead of the function was given - it is important that we have here the product of two functions, each of which depends on "X"Therefore, it is necessary to use a derivation of the product. For the third component, apply the differentiation rule of a complex function.