Cambridge Additional Mathematics

350 Introduction to differential calculus (Chapter 13)

THE CHAIN RULE

If y=g(u) where u=f(x) then

dy

dx

=

dy

du

du

dx

.

This rule is extremely important and enables us to differentiate complicated functions much faster.

For example, for any function f(x):

If y=[f(x)]n then

dy

dx

=n[f(x)]n¡^1 £f^0 (x).

Example 10 Self Tutor

Find

dy

dx

if:

a y=(x^2 ¡ 2 x)^4 b y=

4

p

1 ¡ 2 x

a y=(x^2 ¡ 2 x)^4

) y=u^4 where u=x^2 ¡ 2 x

Now

dy

dx

=

dy

du

du

dx

fchain ruleg

=4u^3 ( 2 x¡ 2 )

=4(x^2 ¡ 2 x)^3 (2x¡2)

b y=

4

p

1 ¡ 2 x

) y=4u

¡^12

where u=1¡ 2 x

Now

dy

dx

=

dy

du

du

dx

fchain ruleg

=4£(¡^12 u

¡^32

)£(¡2)

=4u

¡^32

= 4(1¡ 2 x)

¡^32

EXERCISE 13F.2

1 Write in the form aun, clearly stating whatuis:

a

1

(2x¡1)^2

b

p

x^2 ¡ 3 x c

2

p

2 ¡x^2

d^3

p

x^3 ¡x^2 e^4

(3¡x)^3

f^10

x^2 ¡ 3

2 Find the gradient function

dy

dx

for:

a y=(4x¡5)^2 b y=

1

5 ¡ 2 x

c y=

p

3 x¡x^2

d y=(1¡ 3 x)^4 e y= 6(5¡x)^3 f y=^3

p

2 x^3 ¡x^2

g y=

6

(5x¡4)^2

h y=

4

3 x¡x^2

i y=2

³

x^2 ¡

2

x

́ 3

The brackets around

22 x¡ are essential.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\350CamAdd_13.cdr Tuesday, 7 January 2014 9:53:32 AM BRIAN