3 Differential Calculus
Contents
Section ..................................................................................................................................... Page
3 Differential Calculus ................................................................................................................... 1
3.1 Differentiation from First Principles ......................................................................................... 3
Note 1 – Differentiation from First Principles............................................................................. 3
Definition 2 – Differentiable ....................................................................................................... 4
Example 3 – Differentiation from First Principles ...................................................................... 5
Further Examples ..................................................................................................................... 5
Note 4 – Limits .......................................................................................................................... 6
Note 5 – Two Useful Limits ....................................................................................................... 6
Example 6 – Differentiation from First Principles ...................................................................... 6
Note 7 – Standard Derivatives .................................................................................................. 7
Further Examples ..................................................................................................................... 7
3.2 Sums and Differences Rule .................................................................................................... 8
Definition 1 – Sums and Differences Rule ................................................................................ 8
Example 2 – Sums and Differences Rule ................................................................................. 8
Further Examples ..................................................................................................................... 8
3.3 Chain Rule.............................................................................................................................. 9
Definition 1 – Chain Rule .......................................................................................................... 9
Example 2 – Chain Rule ........................................................................................................... 9
Further Examples ................................................................................................................... 10
3.4 Product Rule ......................................................................................................................... 11
Definition 1 – Product Rule ..................................................................................................... 11
Example 2 – Using the Product Rule ...................................................................................... 11
Example 3 – Using the Product Rule to find the Gradient at a Point ....................................... 11
Further Examples ................................................................................................................... 11
3.5 The Quotient Rule ................................................................................................................ 12
Definition 1 – The Quotient Rule ............................................................................................. 12
Example 2 – Using the Quotient Rule ..................................................................................... 12
Further Examples ................................................................................................................... 12
3.6 Derivatives of sec x, cosec x, cot x and tan x ....................................................................... 13
Definition 1 – The derivative of sec x ...................................................................................... 13
Definition 2 – The derivative of cosec x .................................................................................. 13
Definition 3 – The derivative of tan x ....................................................................................... 14
Definition 4 – The derivative of cot x ....................................................................................... 14
Page i
Contents
Note 5 – Derivatives of Trigonometric Functions .................................................................... 15
Example 6 – Derivatives of Trigonometric Functions .............................................................. 15
Further Examples ................................................................................................................... 15
3.7 The Derivatives of Exponential and Logarithmic Functions .................................................. 16
Definition 1 – The Derivative of ex........................................................................................... 16
Definition 2 – The Derivative of ln x ........................................................................................ 16
Example 3 – Finding the Derivatives of Exponential and Logarithmic Functions .................... 17
Further Examples ................................................................................................................... 17
3.8 Higher Derivatives ................................................................................................................ 18
Note 1 – Higher Derivatives .................................................................................................... 18
Example 2 – Higher Derivatives .............................................................................................. 19
Note 3 – Continuity of Higher Derivatives ............................................................................... 19
Further Examples ................................................................................................................... 19
Page ii
3 Differential Calculus
3.1 Differentiation from First Principles
3.1 Differentiation from First Principles
Note 1 – Differentiation from First Principles
Differentiation is an important tool in maths used to find the gradient of a tangent to a curve for a
given point. An approximation for the gradient of a curve is to use the gradient of a straight line
y  y1
m 2
where m is the gradient of the line and the points ( x1, y1 ) and ( x2 , y 2 ) lie on the curve.
x2  x1
This approximation gives rise to:
m
y 2  y1
x2  x1
f ( x  h)  f ( x )
( x  h)  x
f ( x  h)  f ( x )
m
h
m
y
y
y
f ( x  h)
f ( x  h)
f (x)
f (x)
O
x
xh
x
O
x
xh
x
As h becomes smaller the approximation for the gradient m 
As h  0 the approximation
 f '( x )  lim
h 0
O
x
f ( x  h)  f ( x )
improves.
h
f ( x  h)  f ( x )
 f '( x ) .
h
f ( x  h)  f ( x )
.
h
The definition of the derivative includes the term h which represents a small change in x and is
often referred to as the x increment denoted by δx . Similarly, f ( x  h)  f ( x ) represents a small
change in y and is referred to as the y-increment denoted by δy .
 δy  dy
This gives rise to the notation f '( x )  lim 
  dx .
h 0 δx


Page 3
3 Differential Calculus
3.1 Differentiation from First Principles
Definition 2 – Differentiable
A function f is differentiable at x if the limit f '( x ) exists as the point x is approached from the left
(left differentiable) and the limit f '( x ) exists as the point x is approached from the right (right
differentiable). If both limits exist and they are the same, then the function f is differentiable at x.
y
y
y  tan x
y  x2
O
O
π
2
x
x
The function y  x 2 is not differentiable
The function y  tan x is not differentiable
at O since the left derivative is negative and
at
the right derivative is positive.
therefore the derivative is undefined at
π
2
since there is a break in the graph,
Note: The derivative f ' of a function f is called the derived function.
Page 4
π
2
.
3 Differential Calculus
3.1 Differentiation from First Principles
Example 3 – Differentiation from First Principles
Find the derivative of each function below from first principles.
(a)
f ( x )  6x
(c)
f ( x )  5 x 3  2x 2
(b)
Further Examples
Maths In Action: Book 1
Page 29
Exercise 1A
Page 5
f (x) 
3
x
3 Differential Calculus
3.1 Differentiation from First Principles
Note 4 – Limits
1.
For f ( x )  c where c is a constant, then lim f ( x )  c .
2.
lim kf (h)  k lim f (h) where k is a constant.
3.
lim(f (h)  g(h))  lim f (h)  lim g(h) .
4.
lim(f (h)  g(h))  lim f (h)  lim g(h) .
5.
f (h )
 f (h )  lim
h 0
lim 

for lim g (h)  0 .

h 0 g ( h )
h 0
g (h )

 lim
h 0
h 0
h 0
h0
h0
h 0
h0
h0
h0
h0
Note 5 – Two Useful Limits
For the purposes of finding the derivatives of certain function the following limits are required:
 sin h 
 cos h  1 
lim 
1
lim 
  0.

h

0
h 0
h


 h 
Example 6 – Differentiation from First Principles
Find the derivative of f ( x )  sin5 x from first principles.
Page 6
3 Differential Calculus
3.2 Sums and Differences Rule
Note 7 – Standard Derivatives
Generally, when differentiating a function it is simpler to use standard derivatives instead of
differentiating from first principles.
Three standard derivatives are:
1.
d
(ax n )  nax n 1
dx
2.
d
(sin ax )  a cos ax
dx
3.
d
(cos ax )  a sin ax
dx
Further Examples
Maths In Action: Book 1
Page 29
Exercise 1B
Page 7
3 Differential Calculus
3.2 Sums and Differences Rule
3.2 Sums and Differences Rule
Definition 1 – Sums and Differences Rule
For the functions f and g which are differentiable let k ( x )  f ( x )  g( x ) , then k '( x )  f '( x )  g '( x ) .
Proof
k(x )  f (x )  g(x )
 f ( x  h)  f ( x ) g ( x  h)  g ( x ) 
k '( x )  lim 


h 0
h
h


 f ( x  h)  f ( x ) 
 g ( x  h)  g ( x ) 
 lim 
 lim 


h 0
h
h

 h 0 

 f '( x )  g '( x )
Example 2 – Sums and Differences Rule
(a)
Differentiate x 4  cos3x .
(c)
For y 
(b)
3 x 2  2x
dy
find
.
2
x
dx
Further Examples
Maths In Action: Book 1
Page 30
Exercise 2
Page 8
For f ( x )  x 3  6 x 2 find f '( x ) .
3 Differential Calculus
3.3 Chain Rule
3.3 Chain Rule
Definition 1 – Chain Rule
For g differentiable at x and f differentiable at g ( x ) , k is differentiable and k '( x )  f '(g( x )).g '( x ) for
k  f (g ( x )) .
Proof
Let u  g ( x ) and t  g( x  h)  g( x ) noting that t  0 as h  0 . Hence g( x  h)  g( x )  t  u  t .
f (g ( x  h ))  f (g ( x )) f (u  t )  f (u )

h
h
f (u  t )  f (u ) t


t
h
 f (g ( x  h ))  f (g ( x )) 
k '( x )  lim 

h 0
h


 f (u  t )  f (u ) t 
 lim 
 
h 0
t
h

 f (u  t )  f (u ) 
t 
 lim 
 lim  

h 0
t

 h 0  h 
 f (u  t )  f (u ) 
 g ( x  h)  g ( x ) 
 lim 
 lim 


h 0
t
h

 h 0 

 f '(u )  g '( x )
 f '(g ( x ))  g '( x )
Example 2 – Chain Rule
(a)
Find the derivative of (3 x  4)10 .
(b)
Page 9
For f ( x )  cos3 x find f '( x ) .
3 Differential Calculus
3.3 Chain Rule
(c)
For y  sin x find
dy
.
dx
(d)
Further Examples
Maths In Action: Book 1
Page 32
Exercise 3A
Page 10
For y 
5
dy
find
.
2
dx
(1  6 x )
3 Differential Calculus
3.4 Product Rule
3.4 Product Rule
Definition 1 – Product Rule
For the functions f and g which are differentiable and k ( x )  f ( x ).g( x ) then
k '( x )  f '( x )g( x )  f ( x )g '( x ) .
Example 2 – Using the Product Rule
Find the derivative of the following functions.
(a)
x 4 cos x
(b)
( x  2)2 (2x 3  1)2
Example 3 – Using the Product Rule to find the Gradient at a Point
For f ( x )  x 2 cos x 2 , find f '( π3 ) .
Further Examples
Maths In Action
Page 35
Exercise 4A/4B
Page 11
3 Differential Calculus
3.5 The Quotient Rule
3.5 The Quotient Rule
Definition 1 – The Quotient Rule
For the functions f and g which are differentiable and k ( x ) 
k '( x ) 
f '( x )g ( x )  f ( x )g '( x )
.
(g ( x ))2
Example 2 – Using the Quotient Rule
Find the derivatives of the following functions.
(a)
x8
x2  7
(b)
x 1
( x  3)3 4
2
Further Examples
Maths In Action
Page 37
Exercise 5A/5B
Page 12
f (x)
then
g( x )
3 Differential Calculus
3.6 Derivatives of sec x, cosec x, cot x and tan x
3.6 Derivatives of sec x, cosec x, cot x and tan x
Definition 1 – The derivative of sec x
The secant function is the reciprocal of the cosine
1
function sec x 
and has the graph shown.
cos x
The derivative of sec x can be obtained through the
quotient rule:
d
d  1 
(sec x ) 
dx
dx  cos x 
0  cos x  1 (  sin x )

cos2 x
sin x

cos2 x
sin x
1


cos x cos x
 tan x sec x
y
1
O
1
π
2
x
3π
2
Definition 2 – The derivative of cosec x
The cosecant function is the reciprocal of the sine
1
function csc x 
and has the graph shown.
sin x
The derivative of csc x can be obtained through the
quotient rule:
d
d  1 
(csc x ) 
dx
dx  sin x 
0  sin x  1 (cos x )

sin2 x
 cos x

sin2 x
cos x
1


sin x sin x
1

 csc x
tan x
  cot x csc x
Page 13
y
1
O
1
π
2π
x
3 Differential Calculus
3.6 Derivatives of sec x, cosec x, cot x and tan x
Definition 3 – The derivative of tan x
y
The tangent function has the graph shown.
The derivative of tan x can be obtained through the
quotient rule:
d
d  sin x 
(tan x ) 
dx
dx  cos x 
cos cos x  sin (  sin x )

cos2 x
cos2 x  sin2 x

cos2 x
1

cos2 x
 sec 2 x
O
π
2
π
3π
2
x
2π
Definition 4 – The derivative of cot x
The cotangent function is the reciprocal of the tangent
1
function cot x 
and has the graph shown.
tan x
The derivative of cot x can be obtained through the
quotient rule:
d
d  1 
(cot x ) 
dx
dx  tan x 
0  tan x  1 (sec 2 x )

tan2 x
 sec 2 x

tan2 x
1
  sec 2 x 
tan2 x
1
cos2 x


cos2 x sin2 x
1
 2
sin x
  csc 2 x
Page 14
y
O
π
2
π
3π
2
2π
x
3 Differential Calculus
3.6 Derivatives of sec x, cosec x, cot x and tan x
Note 5 – Derivatives of Trigonometric Functions
d
sin x  cos x
dx
d
cos x   sin x
dx
d
tan x  sec 2 x
dx
d
csc x   csc x cot x
dx
d
sec x  sec x tan x
dx
d
cot x   csc 2 x
dx
Example 6 – Derivatives of Trigonometric Functions
Find the derivatives of the following functions:
(a)
cot(5 x )
(b)
sec x csc x
(c)
cot 2 (3 x )
Further Examples
Maths In Action
Page 40
Exercise 7
Page 15
3 Differential Calculus
3.7 The Derivatives of Exponential and Logarithmic Functions
3.7 The Derivatives of Exponential and Logarithmic Functions
Definition 1 – The Derivative of ex
The derivative of e x (also written as exp( x ) ) can be obtained from first principles. Let f ( x )  a x
then
 f ( x  h)  f ( x ) 
f '( x )  lim 

h 0
h


 a x h  a x 
 lim 

h 0
h


 a x (a h  1) 
 lim 

h 0
h


 a h  1
 lim a x   lim 

h 0
h 0
 h 
 a h  1
 a x  lim 

h 0
 h 
 ah  1
 eh  1
When a  e then lim 
  lim 
  1.
h 0
 h  h 0  h 
Therefore when f ( x )  e x then f '( x )  e x .
d x
e  ex
dx
generally 
d x
a  a x ln a
dx
Definition 2 – The Derivative of ln x
The derivative of the natural logarithm ln x (or loge x ) can be obtained via the chain rule. Since
eln x  x then
deln x
dx
ln x
e .d ln x
dx
d ln x
dx
d ln x
dx

dx
dx
1
1
eln x
1

x

d
1
ln x 
dx
x
generally 
Page 16
d
1
loga x 
dx
x ln a
3 Differential Calculus
3.7 The Derivatives of Exponential and Logarithmic Functions
Example 3 – Finding the Derivatives of Exponential and Logarithmic Functions
Differentiate
3
(a)
e9x
(c)
4ln(2 x  1)
e(3 x 2)
(d)
5x
(b)
ln(sin x )
(e)
log6 x
Further Examples
Maths In Action
Page 43
Exercise 8A/8B
Page 17
3 Differential Calculus
3.8 Higher Derivatives
3.8 Higher Derivatives
Note 1 – Higher Derivatives
Some functions can be differentiated more than once. The notation used for these higher
derivatives is as follows:
Function
1st Derivative
2nd Derivative
...
nth Derivative
f
f'
f ''
...
f (n)
y
dy
dx
d 2y
dx 2
...
dny
dx n
Continual differentiation of a polynomial will eventually lead to 0 e.g.
y
dy
dx
d 2y
dx 2
d 3y
dx 3
d 4y
dx 4
 x 3  6 x 2  4 x  12
 3 x 2  12 x  4
 6 x  12
6
0
Continual differentiation of a trigonometric function follows a pattern e.g.
f ( x )  cos x
f '( x )   sin x
(  cos( x  π2 ))
f ''( x )   cos x
(  cos( x  π2  π2 )  cos( x  π ))
f '''( x )  sin x
(  cos( x  π2  π2  π2 )  cos( x  32π ))
f ''''( x )  cos x
(  cos( x  π2  π2  π2  π2 )  cos( x  2π ))

f
(n)
( x )  cos( x  nπ2 )
Page 18
3 Differential Calculus
3.8 Higher Derivatives
Example 2 – Higher Derivatives
Find the fourth derivative of y  e x cos x .
Note 3 – Continuity of Higher Derivatives
For a function f which is continuous it is not necessarily true that all of the derivatives of f are
continuous.
Further Examples
Maths In Action
Page 46
Exercise 9A/9B
Page 19