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