4.8 Derivative of A Composite Function
Definition :
If f and g are two functions defined by y = f(u) and u = g(x) respectively then a function defined by y = f [g(x)] or fog(x) is called a composite function or a function of a function.
The theorem for finding the derivative of a composite function is known as the CHAIN RULE.
Theorem :
If f and g are differentiable and are defined by y = f (u) and
u = g (x), then the composite function y = f [ g (x) ] is differentiable
and we have
Corollary :
If y = f (u), u = g (v) and v = h (x) where f, g and h
are differentiable functions of u, v and x respectively, then
Example 18
Show that
=
Solution :
Let y =
and u = ax + b
\
= a
Then,
y = un
\ =
nu^{n1}
=
n(ax + b) n1
Now
by chain rule.
\
= n(ax + b) ^{n1 } x a
= an(ax + b) n1
Example 19
Find
if y = (2x^{3} – 5x^{2} + 4)5
Solution :
Let y = u^{5}
and u = 2x^{3} – 5x^{2} + 4
\=
6x^{2} – 10x
\ =
5u4
Now
by chain rule.
\
