Go from differential to partial derivative. Partial derivatives and total differential

Consider changing a function when setting an increment to only one of its arguments - x i , and let's call it.

Definition 1.7.Partial derivative functions by argument x i called.

Designations:.

Thus, the partial derivative of a function of several variables is actually defined as the derivative of the function one variable - x i... Therefore, all properties of the derivatives proved for a function of one variable are valid for it.

Comment. In the practical calculation of partial derivatives, we use the usual rules for differentiating a function of one variable, assuming the argument by which the differentiation is carried out to be variables, and the remaining arguments to be constant.

1. z \u003d2x² + 3 xy –12y² + 5 x – 4y +2,

2. z \u003d x y,

Geometric interpretation of partial derivatives of a function of two variables.

Consider the surface equation z \u003d f (x, y) and draw a plane x \u003d const. Select the point on the line of intersection of the plane with the surface M (x, y)... If you set the argument at increment Δ at and consider the point T on the curve with coordinates ( x, y +Δ y, z +Δ y z), then the tangent of the angle formed by the secant MT with the positive direction of the O axis at, will be equal to. Passing to the limit at, we obtain that the partial derivative is equal to the tangent of the angle formed by the tangent to the resulting curve at the point M with a positive direction of the O axis at. Accordingly, the partial derivative is equal to the tangent of the angle with the O axis x tangent to the curve obtained by cutting the surface z \u003d f (x, y) plane y \u003dconst.

Definition 2.1. The complete increment of a function u \u003d f (x, y, z) is called

Definition 2.2. If the increment of the function u \u003d f (x, y, z) at the point (x 0, y 0, z 0) can be represented in the form (2.3), (2.4), then the function is called differentiable at this point, and the expression is called principal linear part of the increment or the full differential of the function under consideration.

Designations: du, df (x 0, y 0, z 0).

Just as in the case of a function of one variable, the differentials of the independent variables are considered to be their arbitrary increments, therefore

Remark 1. Thus, the statement “the function is differentiable” is not equivalent to the statement “the function has partial derivatives” - for differentiability, the continuity of these derivatives at the point under consideration is also required.

4. Tangent plane and normal to the surface. The geometric meaning of the differential.

Let the function z \u003d f (x, y) is differentiable in a neighborhood of the point M (x 0, y 0)... Then its partial derivatives and are the slopes of the tangents to the lines of intersection of the surface z \u003d f (x, y)with planes y \u003d y 0and x \u003d x 0that will be tangent to the surface itself z \u003d f (x, y). Let's compose the equation of the plane passing through these lines. The direction vectors of the tangents have the form (1; 0;) and (0; 1;), therefore, the normal to the plane can be represented as their vector product: n \u003d (-, -, 1). Therefore, the equation of the plane can be written as follows:

where z 0 \u003d .

Definition 4.1.The plane defined by equation (4.1) is called tangent planeto the function graph z \u003d f (x, y) at the point with coordinates (x 0, y 0, z 0).

It follows from formula (2.3) for the case of two variables that the increment of the function fin the vicinity of the point M can be represented as:

Therefore, the difference between the applications of the graph of the function and the tangent plane is infinitely small of a higher order than ρ, at ρ→ 0.

In this case, the differential of the function f looks like:

which corresponds incrementing the tangent plane to the graph of the function... This is geometric meaning differential.

Definition 4.2.Nonzero vector perpendicular to the tangent plane at a point M (x 0, y 0) surfaces z \u003d f (x, y)is called normal to the surface at that point.

As the normal to the surface under consideration, it is convenient to take the vector - n = { , ,-1}.

Practical work No. 2

"Differential function"

Purpose of the lesson: Learn to solve examples and problems on a given topic.

Questions of theory (initial level):

1. Application of derivatives to study functions for extremum.

2. The differential of a function, its geometric and physical meaning.

3. Full differential functions of several variables.

4. The state of the organism as a function of many variables.

5. Approximate calculations.

6. Finding partial derivatives and total differential.

7. Examples of the use of these concepts in pharmacokinetics, microbiology, etc.

(self-study)

1. answer questions on the topic of the lesson;

2. solve examples.

Examples of

Find the differentials of the following functions:

1)	2)	3)
4)	5)	6)
7)	8)	9)
10)	11)	12)
13)	14)	15)
16)	17)	18)
19)	20)

Using derivatives to study functions

The condition for increasing the function y \u003d f (x) on the segment [a, b]

The condition for the decrease of the function y \u003d f (x) on the segment [a, b]

The condition for the maximum of the function y \u003d f (x) at x \u003d a

f "(a) \u003d 0 and f" "(a)<0

If at x \u003d a the derivatives f "(a) \u003d 0 and f" (a) \u003d 0, then it is necessary to investigate f "(x) in the vicinity of the point x \u003d a. The function y \u003d f (x) at x \u003d a has a maximum if the derivative f "(x) changes sign from" + "to" - "when passing through the point x \u003d a, in the case of a minimum - from" - "to" + "If f" (x) does not change sign when passing through point x \u003d a, then at this point the extremum function has no

Differential function.

The differential of the independent variable is equal to its increment:

Differential of function y \u003d f (x)

The differential of the sum (difference) of two functions y \u003d u ± v

The differential of the product of two functions y \u003d uv

Differential of the quotient of two functions y \u003d u / v

dy \u003d (vdu-udv) / v 2

Function increment

Δy \u003d f (x + Δx) - f (x) ≈ dy ≈ f "(x) Δx

where Δx: is the argument increment.

Approximate calculation of the function value:

f (x + Δx) ≈ f (x) + f "(x) Δx

Application of the differential in approximate calculations

The differential is used to calculate the absolute and relative errors in indirect measurements u \u003d f (x, y, z.). The absolute error of the measurement result

du≈Δu≈ | du / dx | Δx + | du / dy | Δy + | du / dz | Δz + ...

Relative error of measurement result

du / u≈Δu / u≈ (| du / dx | Δx + | du / dy | Δy + | du / dz | Δz + ...) / u

DIFFERENTIAL FUNCTION.

Function differential as the main part of the function increment and.The concept of the differential of a function is closely related to the concept of a derivative. Let the function f (x) continuous at given values xand has a derivative

D f / Dx \u003d f ¢ (x) + a (Dx), whence the increment of the function Df \u003d f ¢ (x) Dx + a (Dx) Dx,where a (Dх) ® 0at Dx ® 0... Let us define the order of the infinitesimal f ¢ (x) Dx Dx.:

Therefore, infinitesimal f ¢ (x) Dx and Dx have the same order of smallness, that is, f ¢ (x) Dx \u003d O.

Let us define the order of the infinitesimal a (Dx) Dxin relation to infinitesimal Dx:

Therefore, the infinitesimal a (Dx) Dxhas a higher order of smallness compared to infinitesimal Dx, i.e a (Dx) Dx \u003d o.

Thus, the infinitely small increment Df differentiable function can be represented in the form of two terms: infinitesimal f ¢ (x) Dx of the same order of smallness with Dx and infinitesimal a (Dx) Dxof a higher order of smallness compared to infinitesimal Dx. This means that in equality Df \u003d f ¢ (x) Dx + a (Dx) Dxat Dx® 0 the second term tends to zero "faster" than the first, that is a (Dx) Dx \u003d o.

The first term f ¢ (x) Dx,linear with respect to Dxare called differential function f (x) at the point xand denote dyor df (reads "de ygrek" or "de ef"). So,

dy \u003d df \u003d f ¢ (x) Dx.

The analytical meaning of the differential is that the differential of the function is the main part of the increment of the function Df, linear with respect to the increment of the argument Dx... The differential of a function differs from the increment of a function by an infinitesimal of a higher order of smallness than Dx... Really, Df \u003d f ¢ (x) Dx + a (Dx) Dxor Df \u003d df + a (Dx) Dx . Differential Argument dx is equal to its increment Dx: dx \u003d Dx.

Example.Calculate the value of the differential function f (x) \u003d x 3 + 2x,when x ranges from 1 to 1.1.

Decision.Let's find a general expression for the differential of this function:

Substituting values dx \u003d Dx \u003d 1.1–1 \u003d 0.1and x \u003d 1 into the last formula, we get the desired value of the differential: df½ x \u003d 1; = 0,5.

PRIVATE DERIVATIVES AND DIFFERENTIALS.

Partial derivatives of the first order... The first-order partial derivative of the function z \u003d f (x, y ) by argument x at the point in question (x; y)called the limit

if it exists.

Partial derivative of a function z \u003d f (x, y)by argument x indicated by one of the following symbols:

Similarly, the partial derivative with respect to atdenoted and determined by the formula:

Since the partial derivative is the usual derivative of a function of one argument, it is not difficult to calculate it. To do this, you need to use all the rules of differentiation considered so far, taking into account in each case which of the arguments is taken as a "constant number" and which is the "variable of differentiation".

Comment.To find the partial derivative, for example by the argument x - df / dx, it is enough to find the ordinary derivative of the function f (x, y),assuming the latter as a function of one argument x, and at - constant; to find df / dy - on the contrary.

Example.Find the values \u200b\u200bof the partial derivatives of a function f (x, y) \u003d 2x 2 + y 2at the point P (1; 2).

Decision.Considering f (x, y) function of one argument x and using the rules of differentiation, we find

At the point P (1; 2)derivative value

Considering f (x; y) as a function of one argument y, we find

At the point P (1; 2)derivative value

ASSIGNMENT FOR INDEPENDENT WORK OF THE STUDENT:

Find the differentials of the following functions:

Solve the following tasks:

1. How much will the area of \u200b\u200ba square with side x \u003d 10 cm decrease if the side is reduced by 0.01 cm?

2. The equation of body motion is given: y \u003d t 3/2 + 2t 2, where s is expressed in meters, t is in seconds. Find the path s traversed by the body in t \u003d 1.92 s from the beginning of the movement.

LITERATURE

1. Lobotskaya N.L. Fundamentals of higher mathematics - M .: "Higher school", 1978.C198-226.

2. Bailey N. Mathematics in biology and medicine. Per. from English. M .: "Mir", 1970.

3. Remizov A.N., Isakova N.Kh., Maksina L.G. Collection of problems in medical and biological physics - M .: "High school", 1987. S16-20.

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

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

For example, the partial increment of a function with respect to a variable will be

Partial derivative with respect to the independent variable at the point
from the function is called the limit (if any) of the ratio of the private increment
functions to increment
variable while striving
to zero:

A 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 with, and is not related to the point at which
this derivative is calculated.

The calculation of partial derivatives is nothing new in comparison with 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 us show this with examples.

Example 1. Find Partial Derivatives of a Function
.

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

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

Example 2. Hfind partial derivatives and function
.

Decision. 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 with a power function (
). Differentiating this expression with respect to , 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 a function
at the point
.

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

when differentiating by will be permanent
:

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

Now we calculate the values \u200b\u200bof these derivatives at the point
by substituting concrete values \u200b\u200bof variables in their expressions. As a result, we get:

11. Partial and total function differentials

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

where
,
- an infinitely small value.

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

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

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

where is everyone
, depend on 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, the partial differential by defined like this:

.

Full differential
functions 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.

For sufficiently small for a differentiable function
the approximate equalities hold

,

with which you can make approximate calculations.

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

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

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

For differentials of functions of several variables, all theorems on the properties of differentials, proved for the case of functions of one variable, are true, for example: if and - continuous functions of variables
with continuous partial derivatives with respect to all variables, and and - arbitrary constants, then:

(6)

Transcript

1 LECTURE N Total differential, partial derivatives and higher-order differentials Total differential Partial differentials Higher-order partial derivatives Higher-order differentials 4 Derivatives of compound functions 4 Total differential Partial differentials If the function z \u003d f (,) is differentiable, then its total differential dz is equal to dz \u003d a + B () zz Noticing that A \u003d, B \u003d, we write the formula () in the following form zz dz \u003d + () We extend the concept of the differential of a function to independent variables, setting the differentials of the independent variables equal to their increments: d \u003d; d \u003d After this, the formula for the total differential of the function takes the form zz dz \u003d d + d () d + d Example Let \u003d ln (+) Then dz \u003d d + d \u003d Similarly, if u \u003d f (, n) is a differentiable function of n independent n variables, then du \u003d d (d \u003d) \u003d The expression dz \u003d f (,) d (4) is called the partial differential of the function z \u003d f (,) with respect to the variable; the expression dz \u003d f (,) d (5) is called the partial differential of the function z \u003d f (,) with respect to the variable From formulas (), (4), and (5) it follows that the total differential of a function is the sum of its partial differentials: dz \u003d d z + dz Note that the total increment z of the function z \u003d f (,), generally speaking, is not equal to the sum of partial increments. If at the point (,) the function z \u003d f (,) is differentiable and the differential dz 0 at this point, then its total the increment z \u003d zz + + α (,) + β (,) differs from its linear part dz \u003d zz + only by the sum of the last terms α + β, which at 0 and 0 are infinitesimal of a higher order than the terms of the linear part Therefore at dz 0, the linear part of the increment of the differentiable function is called the principal part of the increment of the function and the approximate formula z dz is used, which will be all the more accurate the smaller in absolute value the increments of the arguments are, 97 Example Calculate approximately arctan (), 0

2 Solution Consider the function f (,) \u003d arctan () Applying the formula f (x 0 + x, y 0 + y) f (x 0, y 0) + dz, we get arctan (+) arctan () + [arctan () ] + [arctan ()] or + + arctan () arctan () () + () Put \u003d, \u003d, then \u003d -0.0, \u003d 0.0 Therefore, (0.0 0.0 arctg) arctan ( ) + (0.0) 0.0 \u003d arctan 0.0 \u003d + 0.0 + () + () π \u003d 0.05 0.0 0.75 4 It can be shown that the error resulting from the application of the approximate formula z dz does not exceed the number \u003d М (+), where М is the largest value of the absolute values \u200b\u200bof the second partial derivatives f (,), f (,), f (,) when changing the arguments from to + and from to + Partial derivatives of higher orders If the function u \u003d f (, z) has a partial derivative with respect to one of the variables in some (open) domain D, then the derived derivative, being itself a function of, z, can, in turn, at some point (0, 0, z 0) have partial derivatives with respect to the same or any other variable For the original function u \u003d f (, z), these derivatives will be partial derivatives of the second order If the first derivative was taken, for example ep, by, then its derivative with respect to, z is denoted as follows: f (0, 0, z0) f (0, 0, z0) f (0, 0, z0) \u003d; \u003d; \u003d or u, u, u z z z The derivatives of the third, fourth, and so on orders are defined in a similar way. Note that the higher-order partial derivative taken over different variables, for example,; is called the mixed partial derivative Example u \u003d 4 z, then u \u003d 4 z; u \u003d 4 z; u z \u003d 4 z; u \u003d z; u \u003d 6 4 z; u zz \u003d 4; u \u003d z; u \u003d z; u z \u003d 4 z; u z \u003d 8 z; u z \u003d 6 4 z; uz \u003d 6 4 z Note that the mixed derivatives taken with respect to the same variables, but in different orders, coincide.This property is not true for all, generally speaking, functions, but it holds in a wide class of functions Theorem Suppose that) the function f (,) is defined in the (open) domain D,) in this domain there exist the first derivatives f and f, as well as second mixed derivatives f and f, and finally,) these last derivatives f and f, as functions and, are continuous in some point (0, 0) of the domain D Then at this point f (0, 0) \u003d f (0, 0) Proof Consider the expression

3 f (0 +, 0 f (0 +, 0) f (0, 0 + f (0, 0) W \u003d, where, are nonzero, for example, positive, and, moreover, are so small that D contains the entire rectangle [0, 0 +; 0, 0 +] We introduce an auxiliary function of: f (, 0 f (, 0) ϕ () \u003d, which in the interval [0, 0 +] due to () has a derivative: ff ϕ (, 0 +) (, 0) () \u003d and, therefore, is continuous With this function f (0 +, 0 f (0 +, 0) f (0, 0 f (0, 0) the expression W, which is equal to W \u003d can be rewritten as: ϕ (0 +) ϕ (0) W \u003d Since for the function ϕ () in the interval [0, 0 +] all the conditions of the Lagrange theorem are satisfied, we can, by the formula of finite increments, transform the expression W f so: W \u003d ϕ (0 + θ, 0 f (0 + θ, 0) (0 + θ) \u003d (0<θ<) Пользуясь существованием второй производной f (,), снова применим формулу конечных приращений, на этот раз к функции от: f (0 +θ,) в промежутке [ 0, 0 +] Получим W=f (0 +θ, 0 +θ), (0<θ <) Но выражение W содержит и, с одной стороны, и и, с другой, одинаковым образом Поэтому, можно поменять их роли и, введя вспомогательную функцию: Ψ()= f (0 +,) f (0,), путем аналогичных рассуждений получить результат: W=f (0 +θ, 0 +θ) (0<θ, θ <) Из сопоставления () и (), находим f (0 +θ, 0 +θ)=f (0 +θ, 0 +θ) Устремив теперь и к нулю, перейдем в этом равенстве к пределу В силу ограниченности множителей θ, θ, θ, θ, аргументы и справа, и слева стремятся к 0, 0 А тогда, в силу (), получим: f (0, 0)=f (0, 0), что и требовалось доказать Таким образом, непрерывные смешанные производные f и f всегда равны Общая теорема о смешанных производных Пусть функция u=f(, n) от переменных определена в открытой n-мерной области D и имеет в этой области всевозможные частные производные до (n-)-го порядка включительно и смешанные производные n-го порядка, причем все эти производные непрерывны в D При этих условиях значение любой n-ой смешанной производной не зависит от того порядка, в котором производятся последовательные дифференцирования Дифференциалы высших порядков Пусть в области D задана непрерывная функция u=f(, х), имеющая непрерывные частные производные первого порядка Тогда, du= d + d + + d

4 We see that du is also some function of, If we assume the existence of continuous second-order partial derivatives for u, then du will have continuous first-order partial derivatives and we can talk about the total differential of this differential du, d (du), which is called differential of the second order (or second differential) of u; it is denoted by du We emphasize that the increments d, d, d in this case are considered constant and remain the same when passing from one differential to the next (moreover, d, d will be zeros) So, du \u003d d (du) \u003d d (d + d + + d) \u003d d () d + d () d + + d () d or du \u003d (d + d + d + + d) d + + (d + d + \u003d d + d + + d + dd + dd + + dd + + Similarly, the third-order differential du is defined, and so on.If the function u has continuous partial derivatives of all orders up to the nth inclusive, then the existence of the nth differential is ensured But the expressions for them become more and more complicated You can simplify the notation Let's take out the "letter u" in the expression of the first differential in brackets Then, the notation will be symbolic: du \u003d (d + d + + d) u; du \u003d (d + d + + d) u; dnnu \u003d (d + d + + d) u, which should be understood as follows: first, the "polynomial" in parentheses is formally raised according to the rules of algebra to a power, then all the resulting terms are "multiplied" by u (which n is appended in the numerators at) , and only after that all symbols return their value as derivatives and differentials ud) du 4 Derivatives of complex functions Let we have a function u \u003d f (, z), defined in the domain D, and each of the variables, z, in turn, is a function variable t in some interval: \u003d ϕ (t), \u003d ψ (t), z \u003d λ (t) Suppose, in addition, when t changes, the points (, z) do not go beyond the domain D Substituting the values, and z in function u, we obtain a complex function: u \u003d f (ϕ (t), ψ (t), λ (t)) Suppose that u has continuous partial derivatives u, u, and uz with respect to, and z, and that t, t and zt exist Then it is possible to prove the existence of the derivative of a complex function and calculate it Give the variable t some increment t, then, and z will receive increments, and z, respectively, and the function u will receive an increment u We represent the increment of the function u in the form: (this can be done, since we assumed the existence of continuous partial derivatives u, u, and uz) u \u003d u + u + uz z + α + β + χ z, where α, β, χ 0 for, z 0 Separate both part of the equality on t, we obtain u z z \u003d u + u + uz + α + β + χ t t t t t t t 4

5 Let us now let the increment of t to zero: then, z will tend to zero, since the functions, z of t are continuous (we assumed the existence of derivatives t, t, zt), and therefore, α, β, χ also tend to zero In the limit we obtain ut \u003d ut + ut + uzzt () We see that under the assumptions made, the derivative of a complex function really exists.If we use the differential notation, then du dd dz () will have the form: \u003d + + () dt dt dt z dt Consider now the case of dependence , z in several variables t: \u003d ϕ (t, v), \u003d ψ (t, v), z \u003d χ (t, v) In addition to the existence and continuity of partial derivatives of the function f (, z), we assume here the existence of derivatives of functions, z in t and v This case does not differ significantly from the one already considered, since when calculating the partial derivative of a function of two variables, we fix one of the variables, and we have a function of only one variable, the formula () will be the same z, a () should be rewritten as: \u003d + + (a) tttztz \u003d + + (b) vvv z v Example u \u003d; \u003d ϕ (t) \u003d t; \u003d ψ (t) \u003d cos t u t \u003d - t + ln t \u003d - t- ln sint 5

Functions of several variables In many questions of geometry, natural science and other disciplines, one has to deal with functions of two three or more variables Examples: Area of \u200b\u200ba triangle S a h where a is the base

13. Partial derivatives of higher orders Let \u003d have and are defined on D O. Functions and are also called first-order partial derivatives of a function or first partial derivatives of a function. and in general

Appendix Definition of the derivative Let and the values \u200b\u200bof the argument, a f) and f) - ((the corresponding values \u200b\u200bof the function f () The difference is called the increment of the argument, and the difference is the increment of the function on the interval,

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1. Basic concepts. Functions of several variables. We study the function of several variables using examples of functions of two and three variables, since all these definitions and the results obtained

2.2.7. Differential application to approximate calculations. The differential of the function y \u003d depends on x and is the main part of the increment x. You can also use the formula: dy d Then the absolute error:

Lecture 9. Derivatives and differentials of higher orders, their properties. Extremum points of the function. Fermat's and Rolle's theorems. Let the function y be differentiable on some segment [b]. In this case, its derivative

5 The point at which F F F or at least one of these derivatives does not exist is called a singular point of the surface.At such a point the surface may not have a tangent plane Definition Normal to the surface

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Lecture 3 FNP, partial derivatives, differential

What is the main thing we learned in the last lecture

We learned what a function of several variables with an argument from Euclidean space is. We studied what the limit and continuity are for such a function

What will we learn in this lecture

Continuing the study of FNP, we will study the partial derivatives and differentials for these functions. We will learn how to write the equation of the tangent plane and the normal to the surface.

Partial derivative, full differential FNP. Relationship between the differentiability of a function and the existence of partial derivatives

For a function of one real variable, after studying the topics "Limits" and "Continuity" (Introduction to mathematical analysis), the derivatives and differentials of the function were studied. Let's move on to considering similar questions for a function of several variables. Note that if all arguments are fixed in the FNP, except one, then the FNP generates a function of one argument, for which the increment, differential and derivative can be considered. We will call them, respectively, the partial increment, the partial differential and the partial derivative. Let's move on to precise definitions.

Definition 10. Let a function of variables be given where - an element of the Euclidean space and the corresponding increments of arguments,,…,. When magnitude, the function is called partial increments. The total increment of a function is a magnitude.

For example, for a function of two variables, where is a point on the plane and the corresponding increments of the arguments, the increments,. In this case, the quantity is the total increments of the function of two variables.

Definition 11. Partial derivative of a function of variables a variable is the limit of the ratio of the partial increment of a function with respect to this variable to the increment of the corresponding argument when it tends to 0.

We write Definition 11 as the formula or in expanded form. (2) For a function of two variables, Definition 11 is written in the form of the formulas , ... From a practical point of view, this definition means that when calculating the partial derivative with respect to one variable, all other variables are fixed and we consider this function as a function of one selected variable. The usual derivative is taken on this variable.

Example 4. For the function where find the partial derivatives and the point at which both partial derivatives are 0.

Decision ... Let's calculate the partial derivatives, and we write the system in the form The solution of this system is two points and .

Let us now consider how the concept of differential is generalized to FNP. Recall that a function of one variable is called differentiable if its increment is represented in the form , while the value is the main part of the increment of the function and is called its differential. The quantity is a function of, has the property that, that is, it is a function that is infinitely small in comparison with. A function of one variable is differentiable at a point if and only if it has a derivative at this point. In this case, the constant and is equal to this derivative, i.e., the differential satisfies the formula .

If the partial increment of the FNP is considered, then only one of the arguments changes, and this partial increment can be considered as an increment of a function of one variable, that is, the same theory works. Consequently, the differentiability condition holds if and only if there is a partial derivative, and in this case the partial differential is determined by the formula .

But what is the total differential of a function of several variables?

Definition 12. Variable function is called differentiable at the point if its increment is represented as. In this case, the main part of the increment is called the FNP differential.

So, the differential of FNP is the value. Let us clarify what we mean by the value , which we will call infinitesimal in comparison with the increments of the arguments ... This is a function that has the property that if all increments, except one, are equal to 0, then the equality ... This essentially means that = = + +…+ .

And how are the conditions for the differentiability of the FNP and the conditions for the existence of partial derivatives of this function related?

Theorem 1. If the function of variables is differentiable at the point , then it has partial derivatives with respect to all variables at this point and at the same time.

Evidence. We write equality for and in the form and divide both sides of the resulting equality by. In the resulting equality, we pass to the limit at. As a result, we get the required equality. The theorem is proved.

Consequence. The differential of the variable function is calculated by the formula . (3)

In example 4, the differential of the function was equal to. Note that the same differential at the point is equal to ... But if we calculate it at a point with increments, then the differential will be equal. Note that, the exact value of the given function at the point is equal, but this same value, approximately calculated using the 1st differential, is equal. We see that by replacing the increment of a function with its differential, we can approximately calculate the values \u200b\u200bof the function.

But will a function of several variables be differentiable at a point if it has partial derivatives at that point? Unlike a function of one variable, the answer to this question is negative. The exact formulation of the relationship is given by the following theorem.

Theorem 2. If the function of variables at the point there are continuous partial derivatives with respect to all variables, then the function is differentiable at this point.

as . In each bracket, only one variable changes, so we can apply the Lagrange finite-increment formula in both cases. The essence of this formula is that for a continuously differentiable function of one variable, the difference between the values \u200b\u200bof the function at two points is equal to the value of the derivative at some intermediate point, multiplied by the distance between the points. Applying this formula to each of the brackets, we get. Due to the continuity of the partial derivatives, the derivative at the point and the derivative at the point differ from the derivatives and at the point by quantities and, tending to 0 as tending to 0. But then, obviously,. The theorem is proved. , and the coordinate. Check that this point belongs to the surface. Write the equation for the tangent plane and the equation for the normal to the surface at the specified point.

Decision. Indeed,. We already calculated in the last lecture the differential of this function at an arbitrary point, at a given point it is equal. Therefore, the equation of the tangent plane can be written in the form or, and the equation of the normal, in the form .