III. THE DERIVATIVE 導數
3.1 導數的意義
1. Average and Instantaneous Rates of Change; Tangent Line
If y  f (x) , then
The average rate of change of y with respect to x over the interval x0 , x1  is the slope
msec of the secant line joining the points x0 , f ( x0 )  and x1 , f ( x1 ) on the graph of f:
msec 
f ( x1 )  f ( x0 )
.
x1  x0
The instantaneous rate of change of y with respect to x at the point x 0 is the slope mtan
of the tangent line to the graph of f at the point x 0 :
mtan  lim
x1  x0
f ( x1 )  f ( x0 )
f ( x0  h )  f ( x0 )
or mtan  lim
h

0
h
x1  x0
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2. Definition: The function f ' ( x) defined by the formula
f ' ( x)  lim
h 0
f ( x  x)  f ( x)
f ( x  h)  f ( x )
or f ' ( x)  lim
x 0
h
x
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is called the derivative of f with respect to x. The domain of f ' ( x) consists of all x for
which the limit exists.
Notation: y ' ,
dy df
d
f (x) ;
,
,
dx dx dx
The value of the derivative of y  f (x) with respect to x at x=a:
f ' (a)  lim
h 0
f ( a  h)  f ( a )
dy
or y' xa or
h
dx
or
xa
d
f ( x)
dx
x a
3. Differentiability: If f ' exists at a particular x, we say the f is differentiable (has a
derivative) at x. The process of calculating a derivative is called
differentiation. Informally stated, the most commonly encountered points
of nondifferentiability are: corners, points of vertical tangency, and points
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of discontinuity.
4. Relationship between differentiability and continuity: If f is differentiable at a point a, then f
is also continuous at a.
3.2 Techniques of Differentiation
1.
d
(c )  0 , if c is any real number.
dx
2. Power rule for positive integers:
3. Constant multiple rule:
d n
x  nx n 1 , if n is a positive integer.
dx
d
du
(cu )  c
, if u is differentiable function of x, and c is a
dx
dx
constant.
4.
d
du dv
(u  v) 

, if u and v are differentiable functions of x.
dx
dx dx
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5. Product rule:
d
du
dv
(uv)  v
u
dx
dx
dx
du
dv
u
d u
6. Quotient rule:
   dx 2 dx
dx  v 
v
v
7. Power rule for negative integers:
d n
x  nx n 1 , if n is a negative integer and x  0 .
dx
8. Higher Derivatives 高階導數: f ( n ) 
dn
f the nth derivative of f: f "  ( f ' )' ,
dx n
f " '  ( f ")' , f ( 4)  ( f " ' )' ,
3.3 Derivatives of Trigonometric Functions 三角函數的導函數
1  cos x
sin x
 1 ; lim
0
x 0
x 0
x
x
1. lim
2.
d
(sin x)  cos x
dx
3.
d
(cos x)   sin x
dx
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4.
d
(tan x)  sec 2 x
dx
5.
d
(sec x)  sec x tan x
dx
6.
d
(cot x)   csc 2 x
dx
7.
d
(csc x)   csc x cot x
dx
3.4 The Chain Rule 鏈鎖律
If g is differentiable at the point x and f is differentiable at the point g (x ) , then the
composition f  g is differentiable at the point x. Moreover, if y  f ( g ( x)) and
u  g (x) , then y  f (u ) and
dy dy du


.
dx du dx
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1. Parametric formula for
dy dy dy / dt

:
dx dx dx / dt
2. Parametric Formula for
d 2 y d 2 y dy ' / dt
:

dx 2
dx 2 dx / dt
x  t  t 2
Example: 
3
y  t  t
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