Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

P1^

5

!^ ‘Differentiate y^ =^ x^2 ’ could mean differentiation with respect to x, or t, or any

other variable. In this book, and others in this series, we have adopted the

convention that, unless otherwise stated, differentiation is with respect to the

variable on the right-hand side of the expression. So when we write ‘differentiate

y = x^2 ’ or simply ‘differentiate x^2 ’, it is to be understood that the differentiation is

with respect to x.

!^ The expression ‘increasing at a rate of’ is generally understood to imply

differentation with respect to time, t.

EXAMPLE 5.20 The radius r cm of a circular ripple made by dropping a stone into a pond is

increasing at a rate of 8 cm s−^1. At what rate is the area A cm^2 enclosed by the

ripple increasing when the radius is 25 cm?

SOLUTION

A = πr^2

d

d

A

r^ =^2 πr

The question is asking for d

d

A

t

, the rate of change of area with respect to time.

Now dd dd dd

d

d

When anddd

d

d

A

t

A

r

r

t

r rt

r rt

A

t

=×

=

==

2

25 8

π.

==× 22 π 58 ×

Now d

d

d

d

d

d

d

d

When andd

d

d

d

A

t

A

r

r

t

r r

t

r r

t

A

t

=×

=

==

2

25 8

π.

==× 22 π 58 ×

     1260 cm^2 s−^1.