5 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