Page 274 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

Notes
dy g x
y f x g x log f x
dx f x

dy g x g ( ) x
i.e., f x f x g x log f x
dx f x

dy
This method of finding is called Logarithmic differentiation.
dx
(ii) Let y a f x
Taking logarithms, we get

log y f x log a

Differentiate w.r.t. x
1 dy
log a f x
y dx

dy f x
y log a f x a log a f x
dx

Example: Differentiate the following w.r.t. x :
x x x
x
1. x x 2. x 3. x

to
. . . x
x
x
4. 3 x 5. x
Solution:
1. Let y x x
Taking logs, we get

log y x log x

Differentiate w.r.t. x

1 dy 1
x log x 1
y dx x

dy
y 1 log x
dx

dy x
i.e., x 1 log z
dx

LOVELY PROFESSIONAL UNIVERSITY 267