Logarithmic Differentiation
In this section, we will learn to differentiate certain special class of functions given in
the form
y = f (x) = [u(x)]v (x)
By taking logarithm (to base e) the above may be rewritten as
log y = v(x) log [u(x)]
Using chain rule we may differentiate this to get
1
1
( )
( )
dy
v x
y dx
u x
⋅
=
⋅ u′(x) + v′(x) log [u(x)]
which implies that
[
]
( )
( )
( ) log
( )
( )
dy
yv x
u x
v x
u x
dx
u x

=
⋅ ′
+ ′
⋅




The main point to be noted in this method is that f(x) and u(x) must always be
positive as otherwise their logarithms are not defined This process of differentiation is
known as logarithms differentiation and is illustrated by the following examples:
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
131
Example 27 Differentiate 
2
2
(
3) (
4)
3
4
5
x
x
x
x
−
+
+
+
 w