FURTHER MATHEMATICS
DIFFERENTIATION
Complete Examination & Revision Study Note
Level: Senior Secondary / WAEC-style Further Mathematics
Topic: Differentiation
Purpose: Learning, revision, and examination preparation
A complete, self-contained reference covering first principles, all core
differentiation rules, applications, worked examples, practice questions,
and a full answer key — built for exam success.
Study Notes Series
,Further Mathematics — Differentiation Study & Revision Notes
Contents
1 Introduction to Differentiation 2
2 Differentiation from First Principles 2
3 The Power Rule 2
4 Constants 3
5 Constant Multiplier Rule 3
6 Sum and Difference Rule 3
7 Differentiating Polynomials 3
8 Differentiating Fractions 4
9 Differentiating Roots 4
10 Exponential Functions 4
11 Logarithmic Functions 5
12 Trigonometric Functions 5
13 The Product Rule 6
14 The Quotient Rule 6
15 The Chain Rule 6
16 Second Derivative 7
17 Stationary Points 7
18 Classification of Stationary Points 7
19 Gradient of a Curve 8
20 Equation of a Tangent 8
21 Equation of a Normal 8
22 Implicit Differentiation 9
23 Parametric Differentiation 9
24 Applications of Differentiation 9
25 Differentiation and Motion 10
26 Increasing and Decreasing Functions 10
27 Optimization 10
28 Common Differentiation Mistakes 10
29 Quick Differentiation Table 11
30 Exam-Ready Checklist 11
31 Practice Questions 13
,Further Mathematics — Differentiation Study & Revision Notes
1 Introduction to Differentiation
Differentiation is the process of finding the rate at which one quantity changes with respect to
another.
If y changes as x changes, differentiation tells us how rapidly y is changing when x has a
particular value.
dy
The derivative of y with respect to x is written as , or equivalently y ′ .
dx
Worked Example
dy
If y = x2 , then = 2x.
dx
dy
Therefore, when x = 3: = 2(3) = 6.
dx
This means that at x = 3, the instantaneous rate of change of y with respect to x is 6.
2 Differentiation from First Principles
The derivative of a function can be defined using first principles. If y = f (x), then:
dy f (x + h) − f (x)
= lim
dx h→0 h
This formula is important because it gives the fundamental meaning of differentiation.
Worked Example
Using first principles, differentiate y = x2 .
Start with f (x) = x2 , so f (x + h) = (x + h)2 .
dy (x + h)2 − x2 x2 + 2xh + h2 − x2 2xh + h2
= lim = lim = lim
dx h→0 h h→0 h h→0 h
Cancel h: = lim (2x + h)
h→0
dy
Hence: = 2x
dx
3 The Power Rule
The most important basic rule is:
d n
(x ) = nxn−1
dx
where n is any real number for which the expression is defined.
Worked Example
dy
1. y = x5 ⇒ = 5x4
dx
dy
2. y = x7 ⇒ = 7x6
dx
dy
3. y = x2 ⇒ = 2x
dx
2 ´
, Further Mathematics — Differentiation Study & Revision Notes
1 dy 3
4. y = 3
= x−3 ⇒ = −3x−4 = − 4
x dx x
4 Constants
The derivative of a constant is zero.
d
(c) = 0
dx
Worked Example
dy
y=8 ⇒ = 0, because 8 does not change as x changes.
dx
5 Constant Multiplier Rule
If a function is multiplied by a constant, the constant remains unchanged.
d d
c f (x) = c f (x)
dx dx
Worked Example
dy
y = 6x4 ⇒ = 6(4x3 ) = 24x3
dx
6 Sum and Difference Rule
Differentiate each term separately.
d
f (x) ± g(x) = f ′ (x) ± g ′ (x)
dx
Worked Example
Differentiate y = 4x5 + 3x2 − 7x + 9.
dy
= 20x4 + 6x − 7 + 0 = 20x4 + 6x − 7
dx
7 Differentiating Polynomials
A polynomial is differentiated term-by-term.
Worked Example
y = 7x4 − 5x3 + 8x2 − 2x + 10
3 ´
DIFFERENTIATION
Complete Examination & Revision Study Note
Level: Senior Secondary / WAEC-style Further Mathematics
Topic: Differentiation
Purpose: Learning, revision, and examination preparation
A complete, self-contained reference covering first principles, all core
differentiation rules, applications, worked examples, practice questions,
and a full answer key — built for exam success.
Study Notes Series
,Further Mathematics — Differentiation Study & Revision Notes
Contents
1 Introduction to Differentiation 2
2 Differentiation from First Principles 2
3 The Power Rule 2
4 Constants 3
5 Constant Multiplier Rule 3
6 Sum and Difference Rule 3
7 Differentiating Polynomials 3
8 Differentiating Fractions 4
9 Differentiating Roots 4
10 Exponential Functions 4
11 Logarithmic Functions 5
12 Trigonometric Functions 5
13 The Product Rule 6
14 The Quotient Rule 6
15 The Chain Rule 6
16 Second Derivative 7
17 Stationary Points 7
18 Classification of Stationary Points 7
19 Gradient of a Curve 8
20 Equation of a Tangent 8
21 Equation of a Normal 8
22 Implicit Differentiation 9
23 Parametric Differentiation 9
24 Applications of Differentiation 9
25 Differentiation and Motion 10
26 Increasing and Decreasing Functions 10
27 Optimization 10
28 Common Differentiation Mistakes 10
29 Quick Differentiation Table 11
30 Exam-Ready Checklist 11
31 Practice Questions 13
,Further Mathematics — Differentiation Study & Revision Notes
1 Introduction to Differentiation
Differentiation is the process of finding the rate at which one quantity changes with respect to
another.
If y changes as x changes, differentiation tells us how rapidly y is changing when x has a
particular value.
dy
The derivative of y with respect to x is written as , or equivalently y ′ .
dx
Worked Example
dy
If y = x2 , then = 2x.
dx
dy
Therefore, when x = 3: = 2(3) = 6.
dx
This means that at x = 3, the instantaneous rate of change of y with respect to x is 6.
2 Differentiation from First Principles
The derivative of a function can be defined using first principles. If y = f (x), then:
dy f (x + h) − f (x)
= lim
dx h→0 h
This formula is important because it gives the fundamental meaning of differentiation.
Worked Example
Using first principles, differentiate y = x2 .
Start with f (x) = x2 , so f (x + h) = (x + h)2 .
dy (x + h)2 − x2 x2 + 2xh + h2 − x2 2xh + h2
= lim = lim = lim
dx h→0 h h→0 h h→0 h
Cancel h: = lim (2x + h)
h→0
dy
Hence: = 2x
dx
3 The Power Rule
The most important basic rule is:
d n
(x ) = nxn−1
dx
where n is any real number for which the expression is defined.
Worked Example
dy
1. y = x5 ⇒ = 5x4
dx
dy
2. y = x7 ⇒ = 7x6
dx
dy
3. y = x2 ⇒ = 2x
dx
2 ´
, Further Mathematics — Differentiation Study & Revision Notes
1 dy 3
4. y = 3
= x−3 ⇒ = −3x−4 = − 4
x dx x
4 Constants
The derivative of a constant is zero.
d
(c) = 0
dx
Worked Example
dy
y=8 ⇒ = 0, because 8 does not change as x changes.
dx
5 Constant Multiplier Rule
If a function is multiplied by a constant, the constant remains unchanged.
d d
c f (x) = c f (x)
dx dx
Worked Example
dy
y = 6x4 ⇒ = 6(4x3 ) = 24x3
dx
6 Sum and Difference Rule
Differentiate each term separately.
d
f (x) ± g(x) = f ′ (x) ± g ′ (x)
dx
Worked Example
Differentiate y = 4x5 + 3x2 − 7x + 9.
dy
= 20x4 + 6x − 7 + 0 = 20x4 + 6x − 7
dx
7 Differentiating Polynomials
A polynomial is differentiated term-by-term.
Worked Example
y = 7x4 − 5x3 + 8x2 − 2x + 10
3 ´
