Chapter 27

Methods of differentiation

27.1 Introduction to calculus

Calculusis a branch of mathematics involving or lead-
ing to calculations dealing with continuously varying
functions – such as velocity and acceleration, rates
of change and maximum and minimum values of
curves.
Calculus has widespread applications in science and
engineering and is used to solve complicated problems
for which algebra alone is insufficient.
Calculus is a subject that falls into two parts:

(i) differential calculus,ordifferentiation,which

is covered in Chapters 27 to 36, and

(ii) integral calculus,orintegration, which is cov-

ered in Chapters 37 to 44.

27.2 The gradient of a curve

If a tangent is drawn at a point P on a curve, then the
gradient of this tangent is said to be thegradient of the
curveatP. In Fig. 27.1, the gradient of the curve atP
is equal to the gradient of the tangentPQ.

(^0) x
Q
P
f(x)
Figure 27.1
0
B
A
E D
C
f(x 2 )
f(x 1 )
f(x)
x 1 x 2 x
Figure 27.2
For the curve shown in Fig. 27.2, let the pointsAand
Bhave co-ordinates(x 1 ,y 1 )and(x 2 ,y 2 ), respectively.
In functional notation,y 1 =f(x 1 )andy 2 =f(x 2 )as
shown.
The gradient of the chordAB

BC
AC

BD−CD
ED

f(x 2 )−f(x 1 )
(x 2 −x 1 )
For the curvef(x)=x^2 shown in Fig. 27.3.
(i) the gradient of chordAB

f( 3 )−f( 1 )
3 − 1

9 − 1
2
= 4
(ii) the gradient of chordAC

f( 2 )−f( 1 )
2 − 1

4 − 1
1
= 3