Partial derivatives of the first order and total differential. Lecture n21

Partial derivatives of a function, if they exist not at one point, but on a certain set, are functions defined on this set. These functions may be continuous and in some cases may also have partial derivatives at various points in their domain.

The partial derivatives of these functions are called second-order partial derivatives or second partial derivatives.

Second order partial derivatives are divided into two groups:

· second partial derivatives of a variable;

· mixed partial derivatives of with respect to variables and.

With subsequent differentiation, third-order partial derivatives can be determined, etc. By similar reasoning, partial derivatives of higher orders are determined and written.

Theorem. If all partial derivatives included in the calculations, considered as functions of their independent variables, are continuous, then the result of partial differentiation does not depend on the sequence of differentiation.

Often there is a need to solve the inverse problem, which consists in determining whether the total differential of a function is an expression of the form, where continuous functions with continuous derivatives of the first order.

The necessary condition for a total differential can be formulated as a theorem, which we accept without proof.

Theorem. In order for a differential expression to be in a domain the total differential of a function defined and differentiable in this domain, it is necessary that in this domain the condition for any pair of independent variables and is identically satisfied.

The problem of calculating the second order total differential of a function can be solved as follows. If the expression of a total differential is also differentiable, then the second total differential (or a second-order total differential) can be considered the expression obtained by applying the differentiation operation to the first total differential, i.e. . Analytical expression for the second total differential has the form:

Taking into account the fact that mixed derivatives do not depend on the order of differentiation, the formula can be grouped and represented as quadratic form:

The matrix of quadratic form is:

Let a superposition of functions defined in and

Defined in. At the same time. Then, if and have continuous partial derivatives up to the second order at the points and, then there is a second full differential complex function of the following form:

As you can see, the second complete differential does not have the property of form invariance. The expression of the second differential of a complex function includes terms of the form that are absent in the formula of the second differential of a simple function.

The construction of partial derivatives of a function of higher orders can be continued by performing sequential differentiation of this function:

Where the indices take values from to, i.e. the order derivative is considered as a first-order partial derivative of the order derivative. Similarly, we can introduce the concept of a complete differential of the order of a function, as a complete differential of the first order from a differential of order: .

In the case of a simple function of two variables, the formula for calculating the total differential of the order of the function is

The use of the differentiation operator allows us to obtain a compact and easy-to-remember form of notation for calculating the total differential of the order of a function, similar to Newton's binomial formula. In the two-dimensional case it has the form.

Let the function be defined in some (open) domain D points
dimensional space, and
– a point in this area, i.e.
D.

Partial function increment of many variables for any variable is the increment that the function will receive if we give an increment to this variable, assuming that all other variables have constant values.

For example, partial increment of a function by variable will

Partial derivative with respect to the independent variable at the point
of a function is called the limit (if it exists) of the partial increment ratio
functions to increment
variable while striving
to zero:

The partial derivative is denoted by one of the symbols:

;
.

Comment. Index below in these notations only indicates which of the variables the derivative is taken, and is not related to at what point
this derivative is calculated.

The calculation of partial derivatives is nothing new compared to the calculation of the ordinary derivative; you just need to remember that when differentiating a function with respect to any variable, all other variables are taken as constants. Let's show this with examples.

Example 1.Find partial derivatives of a function
.

Solution. When calculating the partial derivative of a function
by argument consider the function as a function of only one variable , i.e. we believe that has a fixed value. At fixed function
is a power function of the argument . Using the formula for differentiating a power function, we obtain:

Similarly, when calculating the partial derivative we assume that the value is fixed , and consider the function
How exponential function argument . As a result we get:

Example 2. NIT partial derivatives And functions
.

Solution. When calculating the partial derivative with respect to given function we will consider it as a function of one variable , and expressions containing , will be constant factors, i.e.
acts as a constant coefficient at power function(
). Differentiating this expression by , we get:

Now, on the contrary, the function considered as a function of one variable , while expressions containing , act as a coefficient
(
).Differentiating according to the rules of differentiation of trigonometric functions, we obtain:

Example 3. Calculate partial derivatives of functions
at the point
.

Solution. We first find the partial derivatives of this function at an arbitrary point
its domain of definition. When calculating the partial derivative with respect to we believe that
are permanent.

when differentiating by will be permanent
:

and when calculating partial derivatives with respect to and by , similarly, will be constant, respectively,
And
, i.e.:

Now let's calculate the values of these derivatives at the point
, substituting specific variable values into their expressions. As a result we get:

11. Partial and complete differential functions

If now to the partial increment
apply Lagrange's theorem on finite increments in a variable , then, considering continuous, we obtain the following relations:

Where
,
– an infinitesimal value.

Partial differential function by variable is called the principal linear part of the partial increment
, equal to the product of the partial derivative with respect to this variable and the increment of this variable, and is denoted

Obviously, a partial differential differs from a partial increment by an infinitesimal of higher order.

Full function increment of many variables is called the increment that it will receive when we give an increment to all independent variables, i.e.

where is everyone
, depend and together with them tend to zero.

Under differentials of independent variables agreed to imply arbitrary increments
and designate them
. Thus, the expression for the partial differential will take the form:

For example, partial differential By is defined like this:

Full differential
a function of several variables is called the principal linear part of the total increment
, equal, i.e. the sum of all its partial differentials:

If the function
has continuous partial derivatives

at the point
then she differentiable at a given point.

When small enough for a differentiable function
there are approximate equalities

with which you can make approximate calculations.

Example 4.Find the complete differential of a function
three variables
.

Solution. First of all, we find the partial derivatives:

Noticing that they are continuous for all values
, we find:

For differentials of functions of many variables, all theorems about the properties of differentials, proven for the case of functions of one variable, are true, for example: if And – continuous functions of variables
, having continuous partial derivatives with respect to all variables, and And are arbitrary constants, then:

(6)

To simplify the recording and presentation of the material, we will limit ourselves to the case of functions of two variables. Everything that follows is also true for functions of any number of variables.

Definition. Partial derivative functions z = f(x, y) by independent variable X called derivative

calculated at constant at.

The partial derivative with respect to a variable is determined similarly at.

For partial derivatives the following applies: normal rules and differentiation formulas.

Definition. Product of the partial derivative and the increment of the argument X(y) is called partial differential by variable X(at) functions of two variables z = f(x, y) (symbol: ):

If under the differential of the independent variable dx(dy) understand increment X(at), That

For function z = f(x, y) let's find out the geometric meaning of its frequency derivatives and .

Consider the point, point P 0 (X 0 ,y 0 , z 0) on the surface z = f(x,at) and curve L, which is obtained by cutting the surface with a plane y = y 0 . This curve can be viewed as a graph of a function of one variable z = f(x, y) in the plane y = y 0 . If held at the point R 0 (X 0 , y 0 , z 0) tangent to the curve L, then, according to the geometric meaning of the derivative of a function of one variable , Where a – the angle formed by the tangent with positive direction axes Oh.