Page 260 - DMTH201_Basic Mathematics-1

Unit 9: Differential Calculus

2x 7 x 4 3x 2 8x Notes
(x 3 ) 1 2
dy x ( 2 ) 1 log x
Example: Find , if, y
dx x 2 e x
Solution:
2 x d 2 2 d 2 x
x e x 1 log x x 1 log x x e
dy dx dx
dx 2 x 2
x e
1 2 2 x x
2
2 x
x e x 1 log x 2x x 1 log x x e e 2x
x
2
2 x
x e
1 3 x 2 2 x x 3
2 x
x e x 2x e logx x x 1 e logx e 2x 2 x
x
4 2x
x e
2 x
2 x
x e e x 2x e log x x x 2 1 e x log x
3 2x
x e
9.4 Function of a Function (or Composite Function)

If a function is made up of more than one function then it is called a composite function. A
composite function is denoted by the symbol f(g(x)), f(g(h(x))) etc.

To Find the Derivative of a Composite Function

Chain Rule

To find the derivative of f(g(x)), we use a rule called chain rule.

g
x
x
Let y f ( ( )) , u g ( )
y f ( )andu g ( )
u
x
dy du
By differentiating y w.r.t. u, we get and by differentiating u w.r.t. x, we get .
du dx
dy dy du
dx du dx
This is called the chain rule.
h
g
Similarly, let y f ( ( ( ))), u g ( ( )),v h ( )
x
x
x
h
x
v
u
y f ( ),u g ( ),v h ( )
dy du
By differentiating y w.r.t. u, we get , by differentiating u w.r.t. v, we get and by
du dv
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