
Differentiation 4-31
Example 4.30 Find
dy
dx
for the function
y x
x
= (cos ) .
2
Solution: Given y x
x
= (cos ) .
2
Here, f(x) = cos x and g(x) = x
2
.
We have
dy
dx
f x
d
dx
g x f x
g x
= ( ( )) ( ( ) ln ( )).
( )
Thus,
dy
dx
x
d
dx
x x
x x
d
dx
x x
d
dx
x
x
=
= +
(cos ) ( ln cos )
(cos ) (ln cos ) ln cos
2
2
2
2
(( )
(cos )
cos
( sin ) ln cos ( )
x
x x
x
x x x
x
2
2
2
1
2
= − +
= − +(cos ) ( tan ln cos ).x x x x x
x
2
2
2
Example 4.31 Find
dy
dx
for the function
y x e
x x
= ( ) .
2
Solution: Given
y x e
x x
= ( ) .
2
Let
h x x
x
( ) .=
Then using product rule, we have
dy
dx
h x
d
dx
e e
dh
dx
x x
= +( ) ( ) .
2 2
First, let us find
dh
dx
.
Let
f x x( ) = and g x x( ) = for h(x).
∴ =
= +
=