Derivatives
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
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
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Derivatives
Advanced Level Pure Mathematics
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) .
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2 1
x cos , x 0
Example Let f ( x) x . Find f ' (0) .
0,
x0
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 .
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Derivatives
Advanced Level Pure Mathematics
4.2 DIFFERENTIABILITY
1
x sin , x 0
Example Let f ( x) x . Show that x 0 , f is continuous but not differentiable.
0,
x0
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?
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Derivatives
Advanced Level Pure Mathematics
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 x0
Example Let f ( x) , Find f ' (0) .
x x0
Example Let f ( x) x 3 . Find f ' (0) .
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Advanced Level Pure Mathematics
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 .
x0 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 x0
Hence, f (x) is continuous at x 0 .
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Derivatives
Advanced Level Pure Mathematics
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
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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 .
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Derivatives
Advanced Level Pure Mathematics
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 .
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Derivatives
Advanced Level Pure Mathematics
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
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Derivatives
Advanced Level Pure Mathematics
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) ye x
, where x 0
dy
Example Let sin( xy) xy, find .
dx
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Derivatives
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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 cos1 x =
dx 1 x2
Solution
d 1 d -1
Remark (sin 1 x) = , cos1 x =
dx 1 x2 dx 1 x2
d 1 d -1
tan 1 x = , cot1 x =
dx 1 x2 dx 1 x2
d 1 d -1
sec1 x = , csc1 x =
dx x x2 1 dx x x2 1
d d x
Example* (a) Find ln x (b) Find e
dx dx
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Derivatives
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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
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Derivatives
Advanced Level Pure Mathematics
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
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1
tan
(h) y 3x e x (i) y3 x
1 x
(j) yx x
(k) y4 ln x
d 1
2. By considering the inverse of cos x , show that (cos 1 x)
dx 1 x2
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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
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Derivatives
Advanced Level Pure Mathematics
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 n1 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 .
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( 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)
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Derivatives
Advanced Level Pure Mathematics
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 .
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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 n2 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
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Derivatives
Advanced Level Pure Mathematics
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) .
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Derivatives
Advanced Level Pure Mathematics
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
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Derivatives
Advanced Level Pure Mathematics
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.
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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 ' ( ) .
ba
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) .
ba
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 ' ( )
ba
f (b) f (a)
Remark: 1. The Mean Value Theorem still holds for a b . f ' ( p) .
ba
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
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Derivatives
Advanced Level Pure Mathematics
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 .
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Example Let P ( x) a n x n a n 1 x n 1 a0 be a polynomial with real coefficients.
an a
If n1 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.
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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.
ba
( 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.
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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 x0
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
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