
4-16 Calculus – Differentiation and Integration
=
+ − + −
+ −
→
lim
( ( )) ( ( )) ( ) ( )
( ) ( )
h
f g x h f g x
h
g x h g x
g x h g x
0
=
+ −
+ −
+ −
→
lim
( ( )) ( ( ))
( ) ( )
( ) ( )
h
f g x h f g x
g x h g x
g x h g x
h
0
=
+ −
+ −
+ −
=
→ →
lim
( ( )) ( ( ))
( ) ( )
lim
( ) ( )
h h
f g x h f g x
g x h g x
g x h g x
h
f '
0 0
(( ( )) ( ).g x g' x
Example 4.13 Find the derivative of the function
f x x( ) .= −3 5
2
Solution: We use the chain rule to solve this problem.
We assume that
g x x( ) = and h x x( ) = −3 5
2
so that
g h x
x
( )
.
=
=
( )
= −3 5
2
The chain rule states that
f ' x g' h x h' x( ) ( ) ( )=
( )
We have
g' x x( )
/
=
−
1
1 2
(by power rul ...