Cambridge Additional Mathematics

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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

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