Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

6

The reverse chain rule

EXAMPLE 6.15 Find the area between the curve y = x and the y axis between y =   0    and y = 3.

SOLUTION

A = (^) ∫
3
0 x^ dy
= (^) ∫
3
0 y
(^2) dy

=

y^3

0

3

3









=   9    square units.

EXERCISE 6E Find the area of the region bounded by each of these curves, the y axis and the

lines y = a and y = b.

1  y =  3 x + 1, a = 1, b = 7.   2   y = x–, 2 a = 0, b = 2.

3    y =^3 x, a = 0, b = 2.  4   y = x − 1, a = 0, b = 2.

5  y =  4 x, a = 1, b = 2.   6   y =    3 x − 2, a = −1, b = 1. 

The reverse chain rule

ACTIVITY 6.3 (i)    Use the chain rule to differentiate these.

(a) (x − 2)^4 (b)   (2x + 5)^7

(c)

1

() 21 x−^3 (d) () 18 − x

Since y = x, x = y^2

y

x

3

O

y = x

Figure 6.22

y

x

7

1

O

y = 3x + 1

y

x

2

O

y = x – 2

You can think of the chain rule

as being: ‘the derivative of the

bracket × the derivative of the

inside of the bracket’.