§3    Differentiation

1. Differentiation of a function of one variable

1. Basic Concepts

[ Definition of Derivative and Its Geometric Meaning ]   Let the function y = f ( x ) when the independent variable has a change at the point x , the function y has a corresponding change , then when it tends to zero , the limit of the ratio exists ( a definite finite value ) , then this limit is called the derivative of the function f ( x ) at the point x , denoted as

 Figure 5.1

At this time, the function f ( x ) is said to be differentiable at point x ( or the function f ( x ) is differentiable at point x ) .

Geometrically , the derivative of the function f ( x ) is the slope of the tangent to the curve represented by the function y = f ( x ) at point x , i.e.

=

where α is the angle between the tangent of the curve at point x and the x -axis ( Figure 5.1) .

[ one-sided derivative ]

=

and

=

are called the left and right derivatives of the function f ( x ) at point x , respectively.

The necessary and sufficient conditions for the existence of derivatives are :

=

[ Infinite Derivative ]   If at some point x there is

= ±∞

Then the function f ( x ) is said to have infinite derivative at point x . At this time, the graph of the function y = f ( x ) is perpendicular to the x - axis at the tangent of the point x ( when =

When + , the graph of the function f ( x ) is in the same direction as the y -axis at the positive tangent of the point x , and when = -∞ , the direction is opposite ) .

[ The relationship between differentiability and continuity of functions ]   If the function y = f ( x ) has a derivative at point x , then it must be continuous at point x . Conversely , continuous functions do not necessarily have derivatives , such as

The 1° 　function y = | x | is continuous at the point x = 0 , at the point x = 0, the left derivative = -1, the right derivative = 1, and the derivative does not exist ( Figure 5.2) .

Figure 5.2 Figure 5.3 _

function

y = f ( x )=

Continuous at x = 0 , but no derivative exists around x = 0 ( Figure 5.3) .

2. The basic rules of taking derivatives

[ Four arithmetic derivation formulas ] 　If c is a constant , the function u = u ( x ) has derivatives , then

=0                = c

( 0 )

[ Derivative of composite function ]   If y = f ( u ), u = both have derivatives , then

=

[反函数的导数]  如果函数y=f(x)在点x有不等于零的导数,并且反函数x=f1(y)在点y连续,那末  存在并且等于，即

=

[隐函数的导数]  假定函数F(x,y)连续,并且对于每个自变量都有连续的偏导数,而且，则由

F(x,y)=0

=

[用参数表示的函数的导数]  设方程组

（α<t<β

y=

[用对数求导数法]  求一函数的导数,有时先取其对数较为便利,然后由这函数的对数求其导数。

两边各取对数,

lny=pln(xa)qln(xb)rln(xc)

3.函数的微分与高阶导数

[函数的微分]  若函数y=f(x)的改变量可表为

A(x)dx+o(dx)

dy=A(x)dx

dy=dx

[高阶导数]  函数y=f(x)的高阶导数由下列关系式逐次地定义出来(假设对应的运算都有意义)

=

[高阶微分]  函数y=f(x)的高阶微分由下列公式逐次定义：

=

=

[莱布尼茨公式]  若函数u==n阶导数(可微分n),

[ Higher-Order Derivatives of Composite Functions ]   If the function y = f ( u ), u = has an l -order derivative, then

in the formula

,

[ Derivative table of basic functions ]

f ( x )

f ( x )

c

0

x n

nx n - 1

sh x

ch x

ch x

sh x

th x

cth x

sech x

csch x

Ar sech x

f > 0 , take +

Ar csch x

, x >0

# Arch x=

, x >1

f > 0 take + , f < 0

# Arth x =

( x< 1)

ln ch x

th x

Arcth x=

( x>1)

ln

sech x csch x

[ Table of Higher Derivatives of Simple Functions ]

 f ( x ) m ( m - 1) … ( m - n +1) ( when m is an integer and n > m , =0) Here (2 n +1)!!=(2 n +1)(2 n - 1) ( a > 0) sh x sh x ( n is even ) , ch x ( n is odd ) ch x ch x ( n is even ) , sh x ( n is odd )

4. Numerical derivatives

When a function is given in a graph or table , it is impossible to find its derivative by definition , only numerical derivatives can be found by approximation .

[ Graphical differentiation method ]   is suitable for obtaining derivatives of functions given by graphics , such as known s - t diagrams , seeking diagrams , a - t diagrams, etc. in mechanical design. The basic steps are as follows:

(1)       Translate the original coordinate system Oxy along the negative direction of the y -axis by a distance to obtain the coordinate system ( Figure 5.4).

Figure 5.4

(2) Make a tangent M 1 T 1   through the point M 1 ( x 1 , y 1 ) on the curve y = f ( x ) and make a tangent M 1 T 1 . In the coordinate system , pass the point P ( -1,0) as PQ 1 parallel to M 1 T 1 intersects the y - axis at point Q 1 , then the ordinate of the point Q 1 ( point ) is the derivative . Take the ordinate of Q 1 as the ordinate, x 1 is the abscissa to make a point .

(3)  Take several points M 1 , M 2 , , on the curve y = f ( x ) , and obtain more dense points at the places where the curve is more curved . By imitating the above method , the corresponding points , , , and , are obtained in the coordinate system . Sub-connected into a smooth curve , that is, the graph of the derivative function .

[ Difference quotient formula ]   The following simple approximate formula is often used in practice

, ,…,

in the formula

= ( 1st order difference of function f ( x ) at point a )

( 2nd order difference of function f ( x ) at point a )

………………………………

( k -th order difference of function f ( x ) at point a )

In the numerical table of the function , if there is an error , the deviation of the higher-order difference is large , so it is not appropriate to use the above formula to calculate the higher-order derivative .

[ Determining Numerical Derivatives Using Interpolation Polynomials ]   Assuming that the interpolation polynomial P n ( x ) of the function y = f ( x ) has been found , it can be derived , then by approximation , given by

f ( x )= P n ( x )+ R n ( x )

Omit the remainder , get

and so on . Their remainders are correspondingly , , and so on .

It should be noted that when the interpolating polynomial Pn ( x ) converges to f ( x ) , it does not necessarily converge to f ' ( x ) . Also , as h shrinks , the truncation error decreases , but the rounding error increases , so , The method of reducing the step size may not necessarily achieve the purpose of improving the accuracy . Due to the unreliability of using the interpolation method to calculate the numerical differentiation , during the calculation , special attention should be paid to the error analysis , or other methods should be used .

[ Lagrange formula ]   ( derived from Lagrangian interpolation formula , see Chapter 17 , §2, 3 )

in the formula

( )

[ Markov formula ]   ( derived from Newton's interpolation formula , see Chapter 17 , §2, 2 )

( )

In particular , when t = 0 , we have

[ Isometric formula ]

three point formula

Four point formula

Five point formula

[ Using Cubic Spline Function to Calculate Numerical Derivative ] This method can avoid the   unreliability of using interpolation method to calculate numerical derivative . Chapter 17 , §2, 4 ), when the interpolated function f ( x ) has a fourth-order continuous derivative , and hi = x i +1 - x i0 , as long as S ( x ) converges to f ( x ) ), then the derivativemust converge to , and S ( x ) - f ( x ) = O ( H 4 ) , - = O ( H 3 ), , where H is the maximum value of hi , therefore , the cubic spline function can be directly passed

Find the numerical derivative

=

In the formula , , ( i =0,1,2, ) .

If only the derivative at the sample point x i is obtained , then

=

=

2. Differentiation of Multivariable Functions

[ Partial Derivatives and Their Geometric Meaning ]   Let the binary function

u = f ( x , y )

Δu=f(x+Δx,y)f(x,y)

=

=====

=====

5.5

[偏微分]  多变量函数u=f(x1,x2,…,xn)对其中一个变量(例如x1 )的偏微分为

[可微函数与全微分]  若函数u=f(x,y)的全改变量可写为

=+

=A, =B

=+dy

du=+dy                                 (1)

[复合函数的微分法与全导数]

u=f(x,y),x=(t,s),y=(t,s),

=+

=+

Let u = f ( x 1 , x 2 ,…, x n ), and x 1 , x 2 ,…, x n are all functions of t 1 , t 2 ,…, t m , then

……………………………………

Let u = f ( x , y , z ), and y = ( x , t ), z = ( x , t ), then

=

=

Set u = f ( x 1 , x 2 ,…, x n ), x 1 = x 1 ( t ), x 2 = x 2 ( t ), , then the function u = f ( x 1 , x 2 , ) , the total derivative of

[ Homogeneous function and Euler's formula ]   If the function f ( x , y , z ) satisfies the following relation identically

f ( tx , ty , tz )= f ( x , y , z )

Then f ( x , y , z ) is said to be a homogeneous function of degree k . For this kind of function , as long as it is differentiable , we have

( Eulerian formula )

Note that the degree k of a homogeneous function can be any real number , for example , the function

It is a π -order homogeneous function of the independent variables x and y .

[ Differentiation of Implicit Functions ]   Let F ( x 1 , x 2 ,…, x n , u )=0, then

……………………

( Refer to this section , IV ).

[ Higher-Order Partial Derivatives and Mixed Partial Derivatives ] The second-order partial derivatives of the   function u = f ( x 1 , x 2 ,…, x n ) are , ,…, and , , ,…, the latter is called mixed partial derivatives . The third-order partial derivatives are , ,…, , , ,… . Higher-order partial derivatives can be defined similarly .

The mixed partial derivative of the product of functions has the following formula : Let u be a function of x 1 , x 2 ,..., x n , then

Note that mixed partial derivatives are generally related to the order of derivation , but if two partial derivatives of the same order differ only in the order of derivation , then as long as the two partial derivatives are continuous , they must be equal to each other . For example , if At a certain point ( x , y ) the function and both are continuous , then there must be

( x , y )= ( x , y )

[ Higher-Order Total Differentiation ] The second-order total differential of a   binary function u = f ( x , y ) is

d 2 u =d(d u )=

or abbreviated as

d 2 u =

The partial derivative symbols in the formula appear after squaring , , , and they act on the function u = f ( x , y ) , and the following are similar .

The nth -order total differential of the binary function u = f ( x , y ) is

d n u =

The nth -order total differential of a multivariable function u = f ( x 1 , x 2 ,…, x m ) is

d n u =

[ Differential Form of Partial Derivatives ]

( in the table h is the step size in the x -axis direction , and l is the step size in the y -axis direction )

 icon Difference formula icon Difference formula icon Difference formula

3. Function determinant ( or Jacobian ) and its properties

n functions with n arguments

(1)

They are defined in an n -dimensional region D , and have continuous partial derivatives with respect to the independent variable , then the determinant composed of these partial derivatives

It is called the functional determinant or Jacobian of function group (1) . Referred to as

Determinants of functions have a series of properties similar to ordinary derivatives .

1° In addition   to the function group (1) , take the function group defined in the region P and having continuous partial derivatives

Assuming that when the point ( t 1 , t 2 , ) changes in P , the corresponding point ( x 1 , x 2 , ) does not go beyond the area D , then you can pass x 1 , x 2 , y 1 , y 2 , regarded as a composite function of t 1 , t 2 . At this time, we have

= (2)

It is the differential law for compound functions of one variable

y = f ( x ), x = ; =

promotion.

In particular , if t 1 = y 1 , t 2 = y 2 , = y n (in other words , from the new variables x 1 , x 2 , and back to the old variables y 1 , y 2 , ), then It can be obtained by formula (2)

=1

It is the inverse function differentiation rule for unary functions

y = f ( x ), x =   =

promotion.

There are m ( m < n ) functions y 1 , y 2 , with n independent variables x 1 , x 2 , :

where x 1 , x 2 are functions of m independent variables t 1 , t 2 :

Assuming that they all have continuous partial derivatives, then y 1 , y 2 , as functions of t 1 , t 2 , the expression of the functional determinant is

=

The sum on the right-hand side of the equation is taken from all possible combinations of n labels taken m at a time.

When m = 1 , the above formula is the differential formula of the ordinary composite function

Generalization of . Especially when n = 3, m = 2 , there are

4° A system of equations consisting of   n equations with 2 n independent variables

F i ( x 1 , x 2 , ; y 1 , y 2 , )=0 ( i =1,2,…, n )

assumed

0

Consider y 1 , y 2 , as functions of x 1 , x 2 , determined by this equation system , then we have

It is the derivative formula of the implicit function y = f ( x ) determined by F ( x , y )=0

promotion .

The determinant of the 5°   function can be used as a scaling factor for the area ( volume ) .

assumed function

u = u ( x , y ), = ( x , y )

It is continuous on a certain region of the xy plane and has continuous partial derivatives , and it is assumed that on this region

0

Then d u d = d x d y

There are similar expressions for higher dimensional spaces .

Example of the transformation between Cartesian coordinates and spherical coordinates

x = r sin cos , y = r sin sin , z = r cos

The determinant of the function is

= =

Then              d x d y d z = d r d d = d r d d