Partial derivatives partial and total differential. Partial derivatives. Partial and complete differential functions
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 a tangent with the positive direction of the axis Oh.
Or:
Let us similarly fix another variable, i.e. let's cross-section the surface z = f(x, y) plane x = x 0 . Then the function z = f(x 0 , y) can be considered as a function of one variable at:
Where b– the angle formed by the tangent at the point M 0 (X 0 , y 0) with positive axis direction Oy(Fig. 1.2).
Rice. 1.2. Illustration geometric meaning partial derivatives
Example 1.6. Given a function z = x 2 – 3xy – 4at 2 – x + 2y + 1. Find and .
Solution. Considering at as a constant, we get
Counting X constant, we find
Concept of a function of two variables
Magnitude z called function of two independent variables x And y, if each pair of permissible values of these quantities, according to a certain law, corresponds to one completely definite value of the quantity z. Independent Variables x And y called arguments functions.
This functional dependence is analytically denoted
Z = f(x,y),(1)
The values of the arguments x and y that correspond to the actual values of the function z, are considered acceptable, and the set of all admissible pairs of values x and y is called domain of definition functions of two variables.
For a function of several variables, in contrast to a function of one variable, the concepts of its private increments for each of the arguments and concept full increment.
Partial increment Δ x z of the function z=f (x,y) by argument x is the increment that this function receives if its argument x is incremented Δx with constant y:
Δ x z = f (x + Δx, y) -f (x, y), (2)
The partial increment Δ y z of a function z= f (x, y) over the argument y is the increment that this function receives if its argument y receives an increment Δy with x unchanged:
Δ y z= f (x, y + Δy) – f (x, y) , (3)
Full increment Δz functions z=f(x,y) by argument x And y is the increment that a function receives if both of its arguments receive increments:
Δz= f (x+Δx, y+Δy) – f (x, y) , (4)
For sufficiently small increments Δx And Δy function arguments
there is an approximate equality:
Δz Δ x z + Δ y z , (5)
and the smaller it is, the more accurate it is Δx And Δy.
Partial derivatives of a function of two variables
Partial derivative of the function z=f (x, y) with respect to the argument x at the point (x, y) called the limit of the partial increment ratio Δ x z this function to the corresponding increment Δx argument x when striving Δx to 0 and provided that this limit exists:
, (6)
The derivative of the function is determined similarly z=f(x,y) by argument y:
In addition to the indicated notation, partial derivative functions are also denoted by z΄ x , f΄ x (x, y); , z΄ y , f΄ y (x, y).
The main meaning of the partial derivative is as follows: the partial derivative of a function of several variables with respect to any of its arguments characterizes the rate of change of this function when this argument changes.
When calculating the partial derivative of a function of several variables with respect to any argument, all other arguments of this function are considered constant.
Example 1. Find partial derivatives of a function
f (x, y)= x 2 + y 3
Solution. When finding the partial derivative of this function with respect to the argument x, we consider the argument y constant value:
;
When finding the partial derivative with respect to the argument y, the argument x is considered a constant value:
.
Partial and complete differentials of functions of several variables
Partial differential of a function of several variables with respect to which-or from its arguments The product of the partial derivative of this function with respect to a given argument and the differential of this argument is called:
d x z= ,(7)
d y z= (8)
Here d x z And d y z-partial differentials of a function z=f(x,y) by argument x And y. Wherein
dx=Δx; dy=Δy, (9)
Full differential a function of several variables is called the sum of its partial differentials:
dz= d x z + d y z, (10)
Example 2. Let's find the partial and complete differentials of the function f (x, y)= x 2 + y 3 .
Since the partial derivatives of this function were found in Example 1, we obtain
d x z= 2xdx; d y z= 3y 2 dy;
dz= 2xdx + 3y 2 dy
The partial differential of a function of several variables with respect to each of its arguments is main part the corresponding partial increment of the function.
As a result, we can write:
Δ x z d x z, Δ y z d y z, (11)
The analytical meaning of the total differential is that the total differential of a function of several variables represents the main part of the total increment of this function.
Thus, there is an approximate equality
Δz dz, (12)
The use of the total differential in approximate calculations is based on the use of formula (12).
Let's imagine the increment Δz as
f (x + Δx; y + Δy) – f (x, y)
and the total differential is in the form
Then we get:
f (x + Δx, y + Δy) – f (x, y) 
,
, (13)
3.The purpose of students’ activities in class:
The student must know:
1. Definition of a function of two variables.
2. The concept of partial and total increment of a function of two variables.
3. Determination of the partial derivative of a function of several variables.
4. Physical meaning partial derivative of a function of several variables with respect to any of its arguments.
5. Determination of the partial differential of a function of several variables.
6. Determination of the total differential of a function of several variables.
7. Analytical meaning of the total differential.
The student must be able to:
1. Find the partial and total increment of a function of two variables.
2. Calculate partial derivatives of functions of several variables.
3. Find partial and complete differentials of a function of several variables.
4. Use the total differential of a function of several variables in approximate calculations.
Theoretical part:
1. The concept of a function of several variables.
2. Function of two variables. Partial and total increment of a function of two variables.
3. Partial derivative of a function of several variables.
4. Partial differentials of functions of several variables.
5. Full differential functions of several variables.
6. Application of the total differential of a function of several variables in approximate calculations.
Practical part:
1.Find the partial derivatives of the functions:
1) 
; 4) 
;
2) z= e xy+2 x; 5) z= 2tg xe y;
3) z= x 2 sin 2 y; 6) 
.
4. Define the partial derivative of a function with respect to a given argument.
5. What is called the partial and total differential of a function of two variables? How are they related?
6. List of questions to check the final level of knowledge:
1. In the general case of an arbitrary function of several variables, is its total increment equal to the sum of all partial increments?
2. What is the main meaning of the partial derivative of a function of several variables with respect to any of its arguments?
3. What is the analytical meaning of the total differential?
7.Chronocard training session:
1. Organizing time- 5 minutes.
2. Analysis of the topic – 20 min.
3. Solving examples and problems - 40 min.
4. Current knowledge control -30 min.
5. Summing up the lesson – 5 min.
8. List of educational literature for the lesson:
1. Morozov Yu.V. Basics higher mathematics and statistics. M., “Medicine”, 2004, §§ 4.1–4.5.
2. Pavlushkov I.V. and others. Fundamentals of higher mathematics and mathematical statistics. M., "GEOTAR-Media", 2006, § 3.3.
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 is with an argument from Euclidean space. We studied what limit and continuity are for such a function
What will we learn in this lecture?
Continuing our study of FNPs, we will study partial derivatives and differentials for these functions. Let's learn how to write the equation of a tangent plane and a normal to a surface.
Partial derivative, complete differential of the FNP. The connection 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 Calculus), derivatives and differentials of the function were studied. Let us move on to consider similar questions for functions of several variables. Note that if all arguments except one are fixed in the FNP, then the FNP generates a function of one argument, for which increment, differential and derivative can be considered. We will call them partial increment, partial differential and partial derivative, respectively. Let's move on to precise definitions.
Definition 10.
Let a function of variables be given where 
- element of Euclidean space and corresponding increments of arguments , ,…, . When the values are called partial increments of the function. The total increment of a function is the quantity .
For example, for a function of two variables, where is a point on the plane and , the corresponding increments of the arguments, the partial increments will be , . In this case, the value is the total increment of a function of two variables.
Definition 11.
Partial derivative of a function of variables 
over a variable is the limit of the ratio of the partial increment of a function over this variable to the increment of the corresponding argument when it tends to 0.
Let us write Definition 11 as a formula 
or in expanded form. (2) For a function of two variables, Definition 11 will be written in the form of 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 ordinary derivative is taken with respect to this variable.
Example 4. For the function where, find the partial derivatives and the point at which both partial derivatives are equal to 0.
Solution
. Let us calculate the partial derivatives and write the system in the form 
The solution to this system is two points and .
Let us now consider how the concept of differential is generalized to the FNP. Recall that a function of one variable is called differentiable if its increment is represented in the form , and the quantity 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 infinitesimal compared to . A function of one variable is differentiable at a point if and only if it has a derivative at that point. Moreover, the constant and is equal to this derivative, i.e., the formula is valid for the differential.
If a 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, i.e. the same theory works. Therefore, the differentiability condition 
satisfied if and only if the partial derivative exists, in which case partial differential is determined by the formula 
.
What is the total differential of a function of several variables?
Definition 12.
Variable function 
called differentiable at a point 
, if its increment is represented in the form . In this case, the main part of the increment is called the FNP differential.
So, the differential of the FNP is the value. Let us clarify what we mean by quantity 
, which we will call infinitesimal compared to 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 is true 
. Essentially this means that 
= = + +…+ .
How are the conditions for the differentiability of a FNP and the conditions for the existence of partial derivatives of this function related to each other?
Theorem 1.
If a function of variables is differentiable at a point , then it has partial derivatives with respect to all variables at this point and at the same time.
Proof.
We write the equality for and in the form 
and divide both sides of the resulting equality by . In the resulting equality 
let's go to the limit at . As a result, we obtain the required equality. The theorem is proven.
Consequence.
The differential of a function of variables is calculated using 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 to . Note that the exact value of the given function at the point is equal to , but the same value, approximately calculated using the 1st differential, is equal to . We see that by replacing the increment of a function with its differential, we can approximately calculate the values of the function.
Will a function of several variables be differentiable at a point if it has partial derivatives at this 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 a function of variables at a point there are continuous partial derivatives with respect to all variables, then the function is differentiable at this point.
as . Only one variable changes in each bracket, so we can apply the Lagrange finite increment formula in both. The essence of this formula is that for a continuously differentiable function of one variable, the difference in the values of 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 partial derivatives, the derivative at a point and the derivative at a point differ from the derivatives at a point by the quantities and , tending to 0 as , tending to 0. But then, obviously, . The theorem is proven. , and the coordinate. Check that this point belongs to the surface. Write the equation of the tangent plane and the equation of the normal to the surface at the indicated point.
Solution.
Really, . In the last lecture we already calculated the differential of this function at an arbitrary point, at given point it is equal 
. Consequently, the equation of the tangent plane will be written in the form or , and the equation of the normal - in the form 
.
Linearization of a function. Tangent plane and normal to the surface.
Derivatives and differentials of higher orders.
1. Partial derivatives of the FNP *)
Consider the function And = f(P), РÎDÌR n or, what is the same,
And = f(X 1 , X 2 , ..., x n).
Let's fix the values of the variables X 2 , ..., x n, and the variable X 1 let's give increment D X 1 . Then the function And will receive an increment determined by the equality
= f (X 1 +D X 1 , X 2 , ..., x n) – f(X 1 , X 2 , ..., x n).
This increment is called private increment functions And by variable X 1 .
Definition 7.1. Partial derivative function And = f(X 1 , X 2 , ..., x n) by variable X 1 is the limit of the ratio of the partial increment of a function to the increment of the argument D X 1 at D X 1 ® 0 (if this limit exists).
The partial derivative with respect to X 1 characters
Thus, by definition
Partial derivatives with respect to other variables are determined similarly X 2 , ..., x n. From the definition it is clear that the partial derivative of a function with respect to a variable x i is the usual derivative of a function of one variable x i, when other variables are considered constants. Therefore, all previously studied rules and differentiation formulas can be used to find the derivative of a function of several variables.
For example, for the function u = x 3 + 3xy – z 2 we have
Thus, if a function of several variables is given explicitly, then the questions of the existence and finding of its partial derivatives are reduced to the corresponding questions regarding the function of one variable - the one for which it is necessary to determine the derivative.
Let's consider implicitly given function. Let the equation F( x, y) = 0 defines an implicit function of one variable X. Fair
Theorem 7.1.
Let F( x 0 , y 0) = 0 and functions F( x, y), F¢ X(x, y), F¢ at(x, y) are continuous in some neighborhood of the point ( X 0 , at 0), and F¢ at(x 0 , y 0) ¹ 0. Then the function at, given implicitly by the equation F( x, y) = 0, has at the point ( x 0 , y 0) derivative, which is equal to
.
If the conditions of the theorem are satisfied at any point of the region DÌ R 2, then at each point of this region 
.
For example, for the function X 3 –2at 4 + wow+ 1 = 0 we find
Let now the equation F( x, y, z) = 0 defines an implicit function of two variables. Let's find and. Since calculating the derivative with respect to X produced at a fixed (constant) at, then under these conditions the equality F( x, y=const, z) = 0 defines z as a function of one variable X and according to Theorem 7.1 we get
.
Likewise 
.
Thus, for a function of two variables given implicitly by the equation 
, partial derivatives are found using the formulas: ,
"Differential function"
Purpose of the lesson: Learn to solve examples and problems on this topic.
Theory questions (baseline):
1. Application of derivatives to study functions at extremum.
2. Differential of a function, its geometric and physical meaning.
3. Complete differential of a function of several variables.
4. The state of the body as a function of many variables.
5. Approximate calculations.
6. Finding partial derivatives and total differentials.
7. Examples of use the above concepts in pharmacokinetics, microbiology, etc.
1. answer questions on the topic of the lesson;
2. solve examples.
Examples
Find 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
Condition for the function y = f(x) to increase on the interval [a, b]
Condition for the function y=f(x) to decrease on the segment [a, b]
Condition for maximum function y=f(x)at x=a
f"(a)=0 and f"" (a)<0
If at x=a the derivatives f"(a) = 0 and f"(a) = 0, then it is necessary to study f"(x) in the vicinity of the point x = a. The function y=f(x) at x=a has a maximum , if, when passing through the point x = a, the derivative f"(x) changes sign from “+” to “-”, in the case of a minimum - from “-” to “+” If f"(x) does not change sign when passing through point x = a, then at this point the function has no extremum
Function differential.
The differential of an independent variable is equal to its increment:
Differential of the function y=f(x)
Differential of the sum (difference) of two functions y=u±v
Differential of the product of two functions y=uv
Differential of the quotient of two functions y=u/v
dy=(vdu-udv)/v 2
Function increment
Δy = f(x + Δx) - f(x) ≈ dy ≈ f"(x) Δx
where Δx: - argument increment.
Approximate calculation of the function value:
f(x + Δx) ≈ f(x) + f"(x) Δx
Application of differential in approximate calculations
The differential is used to calculate absolute and relative errors in indirect measurements u = f(x, y, z.). Absolute error of 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.
Differential of a function as the main part of the increment of a function
And. Closely related to the concept of derivative is the concept of differential of a function. Let the function f(x) is continuous for given values X and has a derivative 
D f/Dx = f¢(x) + a(Dx), whence the increment of the function Df = f¢(x)Dx + a(Dx)Dx, Where a(Dx) ® 0 at Dх ® 0. Let us determine 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 = O.
Let us determine the order of the infinitesimal a(Dх)Dх relative to infinitesimal Dx:
Therefore, infinitesimal a(Dх)Dх has a higher order of smallness compared to infinitesimal Dx, that is a(Dx)Dx = o.
Thus, the infinitesimal 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(Dх)Dх higher order of smallness compared to infinitesimal Dx. This means that in equality Df=f¢(x)Dx + a(Dx)Dx at Dx® 0 the second term tends to zero “faster” than the first, that is a(Dx)Dx = o.
First term f¢(x)Dx, linear with respect to Dx, called differential function f(x) at the point X and denote dy or df(read “de igrek” or “de ef”). So,
dy = df = f¢(x)Dx.
Analytical meaning of the differential is that the differential of a function is the main part of the increment of the function Df, linear with respect to the argument increment 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=f¢(x)Dx + a(Dx)Dx or Df = df + a(Dx)Dx . Argument differential dx equal to its increment Dx: dx=Dx.
Example. Calculate the differential value of a function f(x) = x 3 + 2x, When X varies from 1 to 1.1.
Solution. Let's find a general expression for the differential of this function:
Substituting values dx=Dx=1.1–1= 0.1 And x = 1 into the last formula, we get the desired value of the differential: df½ x=1; = 0,5.
PARTIAL DERIVATIVES AND DIFFERENTIALS.
First order partial derivatives. First order partial derivative of the function z = f(x,y ) by argument X at the point in question (x;y) called limit
if it exists.
Partial derivative of a function z = f(x, y) by argument X is indicated by one of the following symbols:
Similarly, the partial derivative with respect to at denoted and defined by the formula:
Since the partial derivative is the ordinary derivative of a function of one argument, it is not difficult to calculate. 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 serves as a “differentiation variable”.
Comment. To find the partial derivative, for example, with respect to the argument x – df/dx, it is enough to find the ordinary derivative of the function f(x,y), considering the latter a function of one argument X, A at– constant; to find df/dy- vice versa.
Example. Find the values of partial derivatives of a function f(x,y) = 2x 2 + y 2 at the point P(1;2).
Solution. Counting 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) a function of one argument y, we find
At the point P(1;2) derivative value
TASK FOR STUDENT'S INDEPENDENT WORK:
Find the differentials of the following functions:
Solve the following problems:
1. How much will the area of a square with side x=10 cm decrease if the side is decreased by 0.01 cm?
2. The equation of body motion is given: y=t 3 /2+2t 2, where s is expressed in meters, t is in seconds. Find the path s traveled by the body in t=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.: “Higher School”, 1987. P16-20.