§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) .
[ onesided 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。
[ HigherOrder 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 . Subconnected 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 higherorder difference is large , so it is not appropriate to use the above formula to calculate the higherorder 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 fourthorder continuous derivative , and hi = x _{i}_{ +1}  x _{i} → 0 , as long as S ( x ) converges to f ( x _{)} ), then the derivative_{}_{}_{}_{}_{}must 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(x_{1},x_{2},…,x_{n})的情形.
[偏微分] 多变量函数u=f(x_{1},x_{2},…,x_{n})对其中一个变量(例如x_{1} )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(x_{1},x_{2},…,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 ).
[ HigherOrder Partial Derivatives and Mixed Partial Derivatives ] The secondorder partial derivatives of the function u = f ( x _{1} , x _{2} ,…, x _{n} ) are , ,…, and , , ,…, the latter is called mixed partial derivatives . The thirdorder partial derivatives are , ,…, , , ,… . Higherorder 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 )_{}
[ HigherOrder Total Differentiation ] The secondorder 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 righthand 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_{}_{}_{}_{}_{}_{}