2 Further
Differentiation and
Integration
Contents
Section ..................................................................................................................................... Page
2 Further Differentiation and Integration ..................................................................................... 1
2.1 The Derivatives of sin x and cos x .......................................................................................... 1
Definition 1 – The Derivatives of sin x and cos x ...................................................................... 1
Example 2 – Differentiating sin x and cos x .............................................................................. 1
Example 3 – Finding the Gradient of the Tangent to sin x and cos x ........................................ 2
Example 4 – Finding the Equation of the Tangent to sin x and cos x ....................................... 2
Further Examples ..................................................................................................................... 2
Example 5 – Differentiating sin x and cos x (more practice) ..................................................... 3
Example 6 – Finding the rate of change with sin x and cos x ................................................... 3
Further Examples ..................................................................................................................... 3
2.2 The Chain Rule ...................................................................................................................... 4
Definition 1 – The Chain Rule ................................................................................................... 4
Example 2 – Using the Chain Rule ........................................................................................... 4
Further Examples ..................................................................................................................... 4
Example 3 – Using the Chain Rule (Harder Examples) ............................................................ 5
Further Examples ..................................................................................................................... 5
2.3 The Chain Rule for Trigonometric Functions .......................................................................... 6
Example – Using the Chain Rule with Trigonometric Functions .............................................. 6
Further Examples ..................................................................................................................... 6
2.4 The Chain Rule for Trigonometric Functions (Finding Stationary Points) ............................... 7
Example 1 – Using the Chain Rule with Trigonometric Functions (Finding Stationary Points) . 7
Further Examples ..................................................................................................................... 7
2.5 Integrating
 (ax  b)
n
dx ..................................................................................................... 8
Definition 1 – Integrating  (ax  b)n dx ..................................................................................... 8
Example 2 – Using the Standard Integral  (ax  b)n dx ............................................................ 8
Further Examples ..................................................................................................................... 8
2.6 Integrating cos x and sin x ...................................................................................................... 9
Definition 1 – The Derivatives of sin x and cos x ...................................................................... 9
Example 2 – Integrating cos x and sin x ................................................................................. 10
Example 3 – Integrating cos x and sin x ................................................................................. 10
Further Examples ................................................................................................................... 10
2.7 Definite Integrals of cos x and sin x ...................................................................................... 11
Page i
Contents
Section ..................................................................................................................................... Page
Example 1 – Definite Integrals of sin x and cos x ................................................................... 11
Further Examples ................................................................................................................... 11
Page ii
2 Further Differentiation and Integration
2.1 The Derivatives of sin x and cos x
2.1 The Derivatives of sin x and cos x
Definition 1 – The Derivatives of sin x and cos x
The derivatives of y  sin x and y  cos x are shown below:
y
y
y  cos x
y  sin x
x
O
y
x
O
y
y  cos x
y   sin x
x
O
d
sin x  cos x
dx
x
O
d
cos x   sin x
dx
Note: In calculus, sin x and cos x always mean the sine and cosine of an angle of x radians (not
degrees).
Example 2 – Differentiating sin x and cos x
Differentiate
(a)
y  5cos x
(b)
Page 1
y  4cos x  2sin x  6
2 Further Differentiation and Integration
2.1 The Derivatives of sin x and cos x
Example 3 – Finding the Gradient of the Tangent to sin x and cos x
Find the gradient of the tangent to cos x  sin x at:
(a)
x
π
2
(b)
x
π
4
Example 4 – Finding the Equation of the Tangent to sin x and cos x
Find the equation of the tangent to y  5x  sin x at x  0 .
Further Examples
Maths In Action
Page 221
Exercise 2A
Page 2
2 Further Differentiation and Integration
2.2 The Chain Rule
Example 5 – Differentiating sin x and cos x (more practice)
Find f '( x ) when
(a)
4  x sin x
f (x) 
x
(b)
f (x) 
x  x 2 cos x
x2
Example 6 – Finding the rate of change with sin x and cos x
Find the rate of change of f ( x )  3cos x  2sin x at x 
Further Examples
Maths In Action
Page 222
Exercise 2B
Page 3
3π
.
4
2 Further Differentiation and Integration
2.2 The Chain Rule
2.2 The Chain Rule
Definition 1 – The Chain Rule
The chain rule states that for y  f (g( x )) where y  f (u ) and u  g (x ) , then
dy
d
 f ' (g ( x )) (g ( x )) .
dx
dx
Example 2 – Using the Chain Rule
Differentiate
(a)
( x  5) 3
(c)
f (x) 
2
(8 x  2)
(b)
( 6 x  4) 7
(d)
f ( x )  ( x 3  4)2 3
Further Examples
Maths In Action
Page 225
Exercise 3A
Page 4
2 Further Differentiation and Integration
2.3 The Chain Rule for Trigonometric Functions
Example 3 – Using the Chain Rule (Harder Examples)
Differentiate with respect the relevant variable, expressing your answer with positive indices.
(a)
f (x) 
5
3
(6  2 x )
(b)
Further Examples
Maths In Action
Page 226
Exercise 3B
Page 5
1

f (a )   a  1  
a

2
2 Further Differentiation and Integration
2.3 The Chain Rule for Trigonometric Functions
2.3 The Chain Rule for Trigonometric Functions
Example – Using the Chain Rule with Trigonometric Functions
Differentiate the following functions.
(a)
f ( x )  sin6x
(b)
f ( x )  cos(7  2x )
(c)
f (x) 
3
sin8 x
(d)
f ( x )  5 cos 4 x
(e)
f ( x )  6  sinx
(f)
f (x) 
Further Examples
Maths In Action
Page 227
Exercise 4A
Page 6
1
1

7x
cos x
2 Further Differentiation and Integration
2.4 The Chain Rule for Trigonometric Functions (Finding Stationary Points)
2.4 The Chain Rule for Trigonometric Functions (Finding Stationary Points)
Example 1 – Using the Chain Rule with Trigonometric Functions (Finding Stationary Points)
Find the stationary points for the function f ( x )  (cos x  1)2 and determine their nature for
0  x π .
Further Examples
Maths In Action
Page 228
Exercise 4B
Page 7
2 Further Differentiation and Integration
2.5 Integrating
2.5 Integrating  (ax  b)n dx
Definition 1 – Integrating  (ax  b)n dx
For an expression of the form (ax  b)n where a, b and n are constants with a  0 and n  1
there is a standard integral.
n
 (ax  b) dx 
(ax  b)n 1
c
(n  1)a
Example 2 – Using the Standard Integral  (ax  b)n dx
Integrate.
(a)
(c)
(6 x  4)7

1
1
(b)
(1  x )2 dx
Further Examples
Maths In Action
Page 229
Exercise 5
Page 8
(9  2x )3 4
2 Further Differentiation and Integration
2.6 Integrating cos x and sin x
2.6 Integrating cos x and sin x
Definition 1 – The Derivatives of sin x and cos x
The derivatives of y  sin x and y  cos x are shown below:
y
y
y  cos x
y  sin x
x
O
y
x
O
y
y   cos x
y  sin x
x
O
 sin x dx   cos x  c
x
O
 cos x dx  sin x  c
Generally
 sin(ax  b) dx  
1
a
 cos(ax  b) dx  sin(ax  b)  c
cos(ax  b)  c
Page 9
2 Further Differentiation and Integration
2.6 Integrating cos x and sin x
Example 2 – Integrating cos x and sin x
Integrate the following expressions.
(a)
6sin x
(c)
2sin(3x  4)
(b)
cos 4x
Example 3 – Integrating cos x and sin x
Find the particular solution to the differential equation
Further Examples
Maths In Action
Page 230
Exercise 6
Page 10
dy
π
 5cos2 x when y  4 and x  .
dx
2
2 Further Differentiation and Integration
2.7 Definite Integrals of cos x and sin x
2.7 Definite Integrals of cos x and sin x
Example 1 – Definite Integrals of sin x and cos x
Evaluate
(a)

π 2
(b)

3π 4
π 4
π 4
sin2 x dx
cos  2x  π2  dx
Further Examples
Maths In Action
Page 232
Exercise 7
Page 11