§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

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 _                                                   

    2° 　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=f^－1(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

所决定的函数y=f(x)的导数

=＝

式中＝，＝(见本节，四)。

[用参数表示的函数的导数]  设方程组

　（α<t<β）

式中和为可微分的函数,且,则由隐函数存在定理(本节,四,1)可把y确定为x的单值连续函数

y=

而函数的导数可用公式

＝

求得。

[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。

例  求

的导数。

解  两边各取对数,得

lny=pln(x－a)＋qln(x－b)－rln(x－c)

左边的lny为y的函数,而y又为x的函数,故应用求复合函数的导数的法则得到

由此得

所以

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)

式中dx=Δx，则此改变量的线性主部A(x)dx称为函数y的微分,记作

dy=A(x)dx

函数y=f(x)的微分存在的充分必要条件是:函数存在有限的导数=,这时函数的微分是

dy=dx

上式具有一阶微分的不变性,即当自变量x又是另一自变量t的函数时,上面的公式仍然成立.

[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：

= 

式中.并且有

=

及                                

[莱布尼茨公式]  若函数u=及=有n阶导数(可微分n次),则

式中,,为二项式系数。

同样有

式中                                            ，

更一般地有

式中m，n为正整数。

[ 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 ]

 

[ 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 _i → 0 , 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 )

当变量x有一个改变量Δx而变量y保持不变时,得到一个改变量

Δu=f(x+Δx,y)－f(x,y)

如果当Δx→0时,极限

=

存在,那末这个极限称为函数u=f(x,y)关于变量x的偏导数,记作或,也记作或,即

=====

类似地,可以定义二元函数u=f(x,y)关于变量y的偏导数为

=====

偏导数可以按照单变量函数的微分法则求出,只须对所论变量求导数,其余变量都看作常数.

偏导数的几何意义如下:

二元函数u=f(x,y)表示一曲面,通过曲面上一点M(x,y,u)作一平行于Oxu平面的平面,与曲面有一条交线,就是这条曲线在该点的切线与x轴正向夹角的正切,即=.同样,有= (图5.5).

图5.5

偏导数的定义不难推广到多变量函数u=f(x₁,x₂,…,x_n)的情形.

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

也可记作.

[可微函数与全微分]  若函数u=f(x,y)的全改变量可写为

=+

式中A,B与Δx,Δy无关,,则称函数u=f(x,y)在点(x,y)可微分(或可微),这时函数u=f(x,y)的偏导数,一定存在,而且

=A, =B

改变量Δu的线性主部

=+dy

称为函数u=f(x,y)的全微分,记作

du=+dy                                 (1)

函数在一点可微的充分条件:如果在点(x,y)函数u=f(x,y)的偏导数存在而且连续,那末函数在该点是可微的.

公式(1)具有一阶微分的不变性,即当自变量x,y又是另外两个自变量t,s的函数时,上面的公式仍然成立.

上述结果不难推广到多变量函数u=f(x₁,x₂,…,x_n)的情形.

注意,在一个已知点,偏导数的存在一般说来还不能确定微分的存在.

[复合函数的微分法与全导数]

1° 设u=f(x,y),x=(t,s),y=(t,s),则

=+

=+

2° 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

……………………………………

3° Let u = f ( x , y , z ), and y = ( x , t ), z = ( x , t ), then

=

 

=

4° 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.

2°   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.

3°   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