
Differentiation 4-11
( ) ( ) lim
( )( ) ( )( )
lim
[ ( ) ( )]
f g ' x
f g x h f g x
h
f x h g x h
h
h
+ =
+ + − +
=
+ + +
→
→
0
0
−− +
=
+ + + − −
=
+
→
→
[ ( ) ( )]
lim
( ) ( ) ( ) ( )
lim
[ (
f x g x
h
f x h g x h f x g x
h
f x h
h
h
0
0
)) ( )] [ ( ) ( )]
lim
( ) ( )
lim
( ) (
− + + −
=
+ −
+
+ −
→ →
f x g x h g x
h
f x h f x
h
g x h g
h h0 0
xx
h
f ' x g' x
)
( ) ( ).= +
Similarly, we can prove that (f - g)′(x) = f ′(x) - g′(x).
Example 4.7 Find the derivative of the function f(x) = 5x
4
- 3x
3
+ 8x
+ 5.
Solution: Given f(x) = 5x
4
- 3x
3
+ 8x + 5.
Now,
f ' x
d
dx
x x x
d
dx
x
d
dx
x
d
dx
x
d
dx
( ) ( )
( ) ( ) ( ) ( )
= − + +
= − + +
5 3 8 5
5 3 8 5
4 3
4 3
(byy sum and difference rules)
= − + +5 3 8
4 3
d
dx
x
d
dx
x
d
dx
x
d
dx
(