# Derivative by za187cY

VIEWS: 5 PAGES: 28

• pg 1
```									Derivatives

Calculus I                                     4
Chapter 4               Derivatives

4.1   Introduction                                                 2

4.2   Differentiability                                            4

Continuity and Differentiability                             6

4.3   Rules of Differentiation                                     10

4.5   Higher Derivatives                                           17

4.6   Mean Value Theorem                                           22

Prepared by K. F. Ngai(2003)
Page 1
Derivatives

4.1       INTRODUCTION

Let A( x0 , y 0 ) be a fixed point and P( x, y ) be a variable point on the curve y  f (x) as shown on about
y  y0    f ( x)  f ( x 0 )
figure. Then the slope of the line AP is given by                     or                    . When the variable point P
x  x0         x  x0
moves closer and closer to A along the curve y  f (x) , i.e. x  x0 . the line AP becomes the tangent line of
f ( x)  f ( x 0 )
the curve at the point A. Hence, the slope of the tangent line at the point A is equal to lim                                    .
x  x0        x  x0
This term is defined to be the derivative of f (x) at x  x0 and is usually denoted by f ' ( x0 ) . The
definition of derivative at any point x may be defined as follows.

Definition Let y  f (x) be a function defined on the interval a, b and x0  a, b  .
f (x) is said to be differentiable at x 0 ( or have a derivative at x 0 ) if the limit

f ( x)  f ( x 0 )                                                      dy
lim                         exists. This lime value is denoted by f ' ( x 0 ) or          and is called the
x  x0        x  x0                                                             dx x x0

derivative of f (x) at x 0 .
If f (x) has a derivative at every point x in a, b  , then f (x) is said to be differentiable on
a, b .

Remark        As x  x0 , the difference between x and x 0 is very small, i.e. x  x0 tends to zero. Usually,
this difference is denoted by h or x . Then the derivative at x 0 may be rewritten as
f ( x 0  h)  f ( x 0 )
lim                               . ( First Principle )
h 0                h

Example       Let f ( x)  x 2  1 . Find f ' (2) .

Prepared by K. F. Ngai(2003)
Page 2
Derivatives
 2    1
 x cos , x  0
Example       Let f ( x)        x        . Find f ' (0) .
 0,
         x0

Example       If f ( x)  ln x , find f ' (x) .

Example       Let f be a real-valued function defined on R such that for all real numbers x and y,
f ( x  y)  f ( x)  f ( y) . Suppose f is differentiable at x 0 , where x0  R .
f ( h)
(a)    Find the value of lim           .
h 0   h
(b)    Show that f is differentiable on R and express f ' (x) in terms of x 0 .

Prepared by K. F. Ngai(2003)
Page 3
Derivatives

4.2        DIFFERENTIABILITY

      1
 x sin , x  0
Example       Let f ( x)        x        . Show that x  0 , f is continuous but not differentiable.
 0,
         x0
Solution

f (0  h)  f (0)              f (h)  f( 0 )
By definition, f ' (0)      =    lim                        =   lim
h 0           h               h 0         h
1
h sin
=   lim             h
h 0      h
1
=   lim sin     .
h 0      h
1
Since lim sin       does not exist, f is not differentiable at x  0 .
h 0       h

1

Example       If f ( x )  x , show that x  0 , f is continuous but not differentiable.
3

 x2     , if x  1
Example       Let f ( x)                     .
2 x  1 , if x  1

Find f '  ( 1 ) and f '  ( 1 ) by definition. Does f ' (1) exist? Why?

Prepared by K. F. Ngai(2003)
Page 4
Derivatives

Example       Show that f ( x)  x      is not differentiable at x  0 . Find also the derivative of f (x) when

x  0.

x 2  x   x0
Example       Let f ( x)                  , Find f ' (0) .
 x        x0

Example       Let f ( x)  x 3 . Find f ' (0) .

Prepared by K. F. Ngai(2003)
Page 5
Derivatives

 x2    ,x  c
Example       A function f is defined as f ( x)                 .
ax  b , x  c

Find a, b ( in term of c ) if f ' (c) exists.

CONTINUITY AND DIFFERENTIABILITY

Theorem       Let y  f (x) . If f ' ( x) exist, then x  0 implies that y  0 .
y                                 y
Proof         Since y  (      )x , we have lim y      (       )x
x               x 0             x

  lim
y 

 lim x
 x 0 x  x 0
   ( Since both limit exist )


 f ' ( x) lim x .
x 0

As f ' ( x) exists, f ' ( x) is bounded. Futhermore, lim x  0 and so lim y  0 .
x0               x 0

Theorem       If y  f (x) is differentiable at x0 , then f (x) is continuous at x 0 .

Proof         By Theorem, lim y  0 , i.e. lim [ f ( x0  x)  f ( x0 )]  0 .
x 0             x0

Hence, f (x) is continuous at x 0 .

Prepared by K. F. Ngai(2003)
Page 6
Derivatives
Remark        We should have a clear concept about the difference between
(a) f (x) is well-defined at x 0 .
(b) the limit of f (x) at x 0 exists.
(c) f (x) is continuous at x 0 .
(d) f (x) is differentiable at x 0 .

D      C         L

 x2  a2
            ,0  x  a
       a

Example       Let f ( x)         0      ,x  a .

 a( x  a )
2   2
,x  a

      x2
Show f (x) is continuous at x  a and discuss the continuity of f ' ( x) at x  a .

Example       Prove that if f satisfy f ( x  y)  f ( x) f ( y) x, y  R and f ( x)  1  xg ( x) where

lim g ( x)  1 , then f ' ( x) exist x  R and f ' ( x)  f ( x) . Find f (x).
x 0

Prepared by K. F. Ngai(2003)
Page 7
Derivatives
f '(1 )
Example       Prove that if f satisfies f ( xy)  f ( x)  f ( y) , then f '(x)            , where x  0 and
x
find f (x) .

 1
 x           if x  c, c  0

Example       Find a, b in terms of c for f ' (c) exists where f ( x)  
a  bx 2        if x  c



Example       Let f be a real-valued function such that

f ( x)  f (a)  ( x  a) 2            (x, a  R)

Show that f ' ( x)  0 for all x  R .

Prepared by K. F. Ngai(2003)
Page 8
Derivatives

Exercise 4A                                                        Name:                  ________________________
1. Determine which of following functions are differentiable at x  0 .
       1
 x sin
2
( x  0)
(a)      f ( x)  x   3
(b)    f ( x)         x
 0
           ( x  0)

2.   Find f ' ( x) of the following functions. Also, locate the points at which the function is not differentiable.
 4x     ( x  3)
(a)      f ( x)   2                         (b)    f ( x)  x 3  8
2 x  6 ( x  3)

 x3    ,x 1
3.   Let f ( x)                 .
ax  b , x  1

Find the values of a and b so that f is differentiable at x  1 .

Prepared by K. F. Ngai(2003)
Page 9
Derivatives

4.3        RULES OF DIFFERENTIATION

Composite functions

d         du                      d            du dv              d        dv   du
ku  k                            (u  v)                       uv  u  v
dx        dx                      dx           dx dx              dx       dx   dx

du    dv
uv
d u                             dy dy du                        dy   1
   dx 2 dx                                                   
dx  v     v                     dx du dx                        dx dx
dy

Algebraic functions

d k
x  kxk 1           where k must be independent of x (usually a constant)
dx

Inverse functions (esp.: inverse of trigo func)

dy   1
If     y  f 1 ( x)    the n      
dx df ( y )
dy

Trigonometric functions

d                                 d                               d
sin x  cos x                     cos x   sin x                 tan x  sec2 x
dx                                dx                              dx

d                                 d                               d
sec x  sec x tan x               csc x   csc x cot x           cot x   csc2 x
dx                                dx                              dx

Logarithmic functions

d x                                               d         1
e  ex                                            ln x 
dx                                                dx        x

d x                                               d               1
a  a x ln a                                      log a x 
dx                                                dx           x ln a

Parametric functions (commonly use in Rate of change)
dy
dy dt

dx dx
dt

Prepared by K. F. Ngai(2003)
Page 10
Derivatives
Theorem       Chain Rule
If h  g  f , i.e. h( x)  g ( f ( x)) and f, g are differentiable, then h' ( x)  g ' ( f ( x)) f ' ( x) .

Example       h( x)  tan e sin x ,   h' ( x ) =

=

=      sec2 e sin x (e sin x cos x)

Example       Find the derivatives of the following functions:
x
(a)    y  x ln(1  x)                              (b)    y
1 x2

1
sin
(c)    y  (3x  100 ) 60                                  (d)    ye            x
, where x  0

dy
Example       Let sin( xy)  xy, find            .
dx

Prepared by K. F. Ngai(2003)
Page 11
Derivatives
Example       ( Derivatives of inverse function )
d               1
Prove        (sin 1 x) 
dx              1 x2
Solution      Let y  sin 1 x .         x        =   sin y
dy
1           =   cos y
dx
dy             1
=
dx           cos y
d                          1
                  (sin 1 x) =
dx                       1 x2

d                            -1
Example       Prove            cos1 x            =
dx                         1 x2
Solution

d                               1                        d                  -1
Remark           (sin 1 x)     =                             ,           cos1 x   =
dx                             1 x2                     dx                1 x2
d                            1                           d                  -1
tan 1 x       =                             ,           cot1 x   =
dx                         1 x2                         dx               1 x2

d                                1                       d                   -1
sec1 x        =                             ,           csc1 x   =
dx                         x x2 1                       dx               x x2 1

d                                          d x
Example*      (a) Find         ln x                   (b) Find            e
dx                                         dx

Prepared by K. F. Ngai(2003)
Page 12
Derivatives
dy
Example       Find         if
dx
(a)    y  a x 1 , where a is a constant.
(b)    y  x x  sin x .

Example       Find the derivative of following functions
(a) x 2 sin 3 (1  3x)             (b) (ln (cos 2 x)) 3

dy
Example       If x  cost and y  sin t , find
dx

Prepared by K. F. Ngai(2003)
Page 13
Derivatives

Exercise 4B                                                                 Name     ________________________

1.   Find the derivatives of the following functions.

(a)      y  e sin x               (b)   y  tan e x                      (c)       
y  ln( x 2  1)   
3

sin x
(d)      y  ln(tan e x )                     (e)      y
sin 1 x

 2x 
(f)      y  tan 1     2 
(g)      y  (cos x) x
1 x 

Prepared by K. F. Ngai(2003)
Page 14
Derivatives
1
tan
(h)      y  3x e x                         (i)    y3         x

1                                           x

(j)      yx   x
(k)    y4   ln x

d                1
2.   By considering the inverse of cos x , show that      (cos 1 x) 
dx              1  x2

Prepared by K. F. Ngai(2003)
Page 15
Derivatives
dy
3.   Find         if
dx
1
(a)      tan 1 y  xy  x 2           (b)    x 2  y 2  x cos y  1
y

dy
4.   Find         if
dx

 x  t  cost                      x  cos3 t
(a)                                    (b) 
 y  1  sin t                     y  sin t
3

Prepared by K. F. Ngai(2003)
Page 16
Derivatives

4.5       HIGHER DERIVATIVES

Definition    If y  f (x) is a function of x , then the nth derivatives of y w.r.t. x is defined as

d n y d  d n1 y  d n 1 y
  n 1  if           is differentiable.
dx n dx  dx 
          dx n 1

dny
Symbolically, the nth derivatives of y w.r.t. x is denoted by y ( n ) , f   (n)
( x) or        .
dx n
dy      1             d2y     1
Remark             1.                    but       2

dx      dx            dx     d 2x
dy                   dy 2

d 2 y d  dy 
   
dx 2 dx  dx 

dy dy dz    dy             dz
2.    If y is function of z , z  f (x) , then                (.     f ' ( x)  ).
dx dz dx    dx             dx
d 2 y d  dy                            dy           d dy
=                =   f ' ' ( x)       f ' ( x) ( )
dx 2 dx  dx                            dz           dx dz
dy         dz d  dy 
=   f ''(x)       f '(x)   
dz         dx dz  dz 
2
dy           2 d y
=   f ''(x)  ( f '(x))
dz             dz 2

d2y
Example       Let y  sin 1 x . Find        .
dx 2

Example       Prove that y  e  kx (a cos x  b sin x) satisfies the equation y ' '2ky'(k 2  1) y  0 .

Prepared by K. F. Ngai(2003)
Page 17
Derivatives
( n 1)
Example       If y  xu , where u is a function of x, Prove y             (n)
 xu   (n)
 nu             where y (n ) and u (n ) are
the nth derivative of y and u respectively.
1                                    1

Hence, if y  x n 1e x , prove y ( n )  (1) n x ( n 1) e x .

Example       Find a general formula for the n th derivative of
(a) y  e ax ( a  R )
(b) y  sin x
(c)    y  xm      ( m  R)

Prepared by K. F. Ngai(2003)
Page 18
Derivatives
Theorem       Let f (x) and g (x) be two functions which are both differentiable up to nth order. Then
dn              dn
(a)         kf ( x)  k n f ( x)
dx n            dx
dn                       dn         dn
(b)         [ f ( x)  g ( x)]  n f ( x)  n g ( x)
dx n                     dx         dx

Theorem       Leibniz's Theorem
Let f and g be two functions with nth derivative. Then
dn                        n
[ f ( x) g ( x)]   C rn f   (r )
( x)g ( n  r ) ( x) where f   (0)
( x)  f ( x) .
dx n                    r 0

Example       Let y  xe ax , where a is a real constant. Find y ( 20) .

dn 3
Example       Find        ( x ln x) , x  0 .
dx n

Example       Let y  tan 1 x . Show that for n  1 , (1  x 2 ) y ( n  2)  2(n  1) xy ( n 1)  n(n  1) y ( n )  0 .

Prepared by K. F. Ngai(2003)
Page 19
Derivatives
2
d y       dy
Example       Prove that if y  x 2 cos x, then x 2            2
 4 x  ( x 2  6) y  0.
dx        dx
dny          d n2 y
Deduce that when x  0 (n  2)( n  3) n  n(n  1) n  2  0
dx           dx

(a) Prove if y  e  x , y ( n  2)  2 xy ( n 1)  2(n  1) y ( n )  0 .
2
Example

d n  x2      x2           d2                d
(b) Show if    n
(e )  e f ( x) , then    2
f ( x)  2 x    f ( x)  2nf ( x)  0 .
dx                          dx                dx

Prepared by K. F. Ngai(2003)
Page 20
Derivatives

Exercise 4D                                                                                Name    ____________________

 x  1
r

1.   Let r be a real number. Define y                            for x  1 .
 x 1

dy  2ry
(a) Show that              .
dx x 2  1
(b) For n  1, 2, 3, , show that
( x 2  1) y ( n 1)  2(nx  r ) y ( n )  (n 2  n) y ( n 1)  0
dky
where y ( 0)  y and y ( k )             for k  1 .
dx k

2.   Let f ( x)  x n e ax , where a is real and n is a positive integer.
Evaluate f ( 2 n ) (0) .

Prepared by K. F. Ngai(2003)
Page 21
Derivatives

4.6           MEAN VALUE THEOREM

Definition    Let y  f (x) be a function defined on an interval I. f is said to have an absolute maximum at
c if f (c)  f ( x), x  I and f (c) is called the absolute maximum value.
Similarly, f is said to have an absolute minimum at d if f (d )  f ( x), x  I and f(d) is
called the absolute minimum value.

Theorem       Fermat's Theorem
Let y  f (x) be defined and differentiable on an open interval (a, b). If f (x) attains its
absolute maximum or absolute minimum (both are called absolute extremum) at x  c ,
where c  (a, b) , then f ' (c)  0 .

Proof         For any x  (a, b), there exists a real number h such that ( x  h)  (a, b) and ( x  h)  (a, b) .
Now, suppose f (x) attains its absolute maximum at x  c . Then we have (c  h)  (a,b)
and (c  h)  (a,b) , and so f (c  h)  f (c) and f (c  h)  f (c) . Now, the left and right
hand derivatives are given by
f(c  h)  f(c)
f'(c)  lim                   0,         ( since f(c  h)  f(c)  0 )
h 0           h
f(c  h)  f(c)
and        f'(c)  lim                0.         ( since f(c  h)  f(c)  0 )
h 0        h
Since f (x) is differentiable at x  c , the left and right hand derivatives must be equal,
i.e. f '  (c)  f '  (c) . This is possible only if f ' (c)  0 .
The proof for f (x) attaining its absolute minimum at x  c is similar and is left as an exercise.

Remark        1.    f ' ( p)  0 NOT IMPLIES            absolute max. or min. at x  p .
e.g. f ( x)  x 3 at x  0 , not max. and min.

figure
2.   Fermat's Theorem can't apply to function in closed interval. absolute max. or min may be
attained at the end-points. As a result, one of the left and right hand derivatives at c may
not exist.
e.g. f ( x)  ( x  2) 2  1defined on [ 0, 5] attains its absolute max. at x  5 but its right
hand derivative does not exist.
3.   Fermat' s Theorem can't apply to function which are not differentiable.

e.g. f ( x)  x . Not differentiable at x  0 but min. at x  0 .

figure
Prepared by K. F. Ngai(2003)
Page 22
Derivatives

Theorem       Rolle's Theorem
If a function f (x) satisfies all the following three conditions:
(1) f (x) is continuous on the closed interval [a, b] ,
(2) f (x) is differentiable in the open interval (a, b) ,
(3) f (a)  f (b) ;
then there exists at least a point   (a, b) such that f ' ( )  0 .

Proof         Since f (x) is continuous on [a, b]  f (x) is bounded
(i)   m  M , where m (min), M (Max) are constant.
        m  f ( x)  M , x  [a, b]
     f ( x)  M , x  [a, b]
     f ' ( x)  0, x  (a, b)
(ii) m  M , the max. and min. cannot both occur at the end points a, b.
 p  (a, b) such that f ( p)  M
i.e. f ( p)  f ( x) x sufficiently closed to p.
By Fermat's Theorem, f ' ( p) exist and equal to 0.

Example       Define f ( x)  ( x  2) 2  1 on [0,4]. Note that f (0)  f (4)  5 .
We have f ' ( x)  2( x  2) and so f ' (2)  0 . Since 2  (0,4) , Rolle's Theorem is verified.
The geometric significance of Rolle's theorem is illustrated in the following diagram.

If the line joining the end points (a, f (a)) and (b, f (b)) is horizontal (i.e. parallel to the x-axis)
then there must be at least a point  (or more than one point) lying between a and b such that the
tangent at this point is horizontal.

Prepared by K. F. Ngai(2003)
Page 23
Derivatives
Theorem       Mean Value Theorem
If a function f (x) is
(1) continuous on the closed interval [a, b] and
(2) differentiable in the open interval (a, b) ,
then there exists at least a point   (a, b) such that
f (b)  f (a)
 f ' ( ) .
ba

Proof         Consider the function g defined by
f (b)  f (a)
g ( x)               ( x  a)  f (a)  g (x) is differentiable and continuous on (a, b) .
ba
Let h( x)  g ( x)  f ( x)
 h(x) is also differentiable and continuous on (a, b) .
We have h(a)  0, h(b)  0.
     By Rolle's Theorem,       (a, b) such that h ' ( )  0
 g ' ( )  f ' ( )  0
f (b)  f (a)
      f ' ( ) 
ba
f (b)  f (a)
Remark:       1.    The Mean Value Theorem still holds for a  b . f ' ( p)                .
ba
2.    Another form of Mean Value Theorem f (b)  f (a)  f ' ( p)(b  a)
3.    The value of p can be expressed as p  a   (b  a) , 0    1.
 f (b)  f (a)  f ' (a   (b  a))(b  a)

Example       Use the Mean Value Theorem. show a  b

(a)   sin a  sin b  a  b

cos ax  cos bx
(b)                    a b , x  0
x
sin px
(c)           p,      p  0, x  0 .
x

Prepared by K. F. Ngai(2003)
Page 24
Derivatives
Example       By using Mean Value Theorem, show that
e y  e a  e a ( y  a)
for all real values y and a .

Solution      Let f ( x)  e x .
Case (i) y  a

Case (ii) y  a

Case (iii) y  a

Example       Let a, b  R such that a  b and f (x) be a differentiable function on R such that
f (a)  0 , f (b)  0 and f ' ( x) is strictly decreasing. Show that f ' (b)  0 .

Example       Let f (x) be a continuous function defined on [ 3, 6 ]. If f (x) is differentiable on ( 3, 6 ) and

f ' ( x)  9  3 . Show that 18  f (6)  f (3)  36 .

Prepared by K. F. Ngai(2003)
Page 25
Derivatives

Example       Let P ( x)  a n x n  a n 1 x n 1    a0 be a polynomial with real coefficients.

an     a
If         n1    a0  0, by using Mean Value Theorem, show that the equation P( x)  0 has
n 1      n
at least one real root between 0 and 1.

Example       Let f be a real-valued function defined on (0, ) . If f ' (t ) is an increasing function,
show that     f (n)  f ' (n)(t  n)  f (t )  f (n)  f ' (n  1)(t  n) (t (n, n  1))

Example       Let f be a real-valued function such that

f ( x)  f ( y)  ( x  y) 2 ,   (x, y  R)

Show that f is a differentiable function.
Hence deduce that f ( x)  k for all x  R , where k is a real constant.

Prepared by K. F. Ngai(2003)
Page 26
Derivatives
Example       Let f (x) be a function such that f ' ( x) is strictly increasing for x  0 .
(a) Using Mean Value Theorem, or otherwise, show that
f ' (k )  f (k  1)  f (k )  f ' (k  1) ,                k 1
(b) Hence, deduce that
f ' (1)  f ' (2)    f ' (n  1)  f (n)  f (1)  f ' (2)  f ' (3)   f ' (n) , n  2

Theorem       Generalized Mean Value Theorem
Let f (x) and g (x) such that
(i) f (x) and g (x) are continuous on [ a, b ].
(ii) f (x) and g (x) are differentiable on ( a, b ).
Then there is at least one points p  (a, b) such that
[ f (b)  f (a)] g ' ( p)  [ g (b)  g (a)] f ' ( p) .

Proof         Let h( x)  [ f (b)  f (a)] g ( x)  [ g (b)  g (a)] f ( x) , a  x  b .
         (i) h(x) is continuous on [ a, b ].
(ii) h(x) is differentiable on ( a, b ).
h(b)  h(a)
By Mean Value Theorem, p  (a, b) such that h ' ( p)                            , hence the result is obtained.
ba
( Why ? )

Remark:       Suppose that f (x) and g (x) are differentiable on ( a, b ) and that f ' ( x)  g ' ( x)  0 ,

f (b)  f (a)   f ' ( p)
x  (a, b) then                              .
g (b)  g (a)   g ' ( p)

This is useful to establish an inequality by using generalized mean value theorem.

Prepared by K. F. Ngai(2003)
Page 27
Derivatives
Example       (a) Let f and g be real-valued functions continuous on [a, b] and differentiable in (a, b) .

(i)   By considering the function

h( x)  f ( x)[g (b)  g (a)]  g ( x)[ f (b)  f (a)] on [a, b] , or otherwise,

show that there is c  (a, b) such that f ' (c)[g (b)  g (a)]  g ' (c)[ f (b)  f (a)]

(ii) Suppose g ' ( x)  0 for all x  (a, b) . Show that g ( x)  g (a)  0 for any x  (a, b) .
f ' ( x)                                              f ( x)  f (a)
If, in addition,            is increasing on (a, b) , show that P ( x)                  is also
g ' ( x)                                              g ( x)  g (a )
Increasing on (a, b) .
 e x cos x  1                   
                   if       x   0, 
 4
 sin x  cos x  1                  
(b) Let Q( x)  
         1         if          x0


 
Show that Q is continuous at x  0 and increasing on 0,  .
 4
       x
Hence or otherwise, deduce that for x  0,  ,  Q (t )dt  x .
 4     0

Prepared by K. F. Ngai(2003)
Page 28

```
To top