Page 292 - DMTH201_Basic Mathematics-1

Unit 10: Logarithmic Differentiation

(ii) The proportional rate of growth is Notes
d log X dr t 2
r b t log a log b 0 b log a log b 0.
dt dt

Hence proportional rate of growth is positive but declines over time.

10.3 Summary

x
x
: f x cos( ) , which cannot be treated as a power g where g x cos( ) or as an
:
x
n
x
exponent e .
We cannot apply the exponential or power rule for differentiating f.
Using the properties of the natural logarithm (ln), we can "simplify" some functions to
allow us to apply the product rule, and logarithmic rule for differentiating
d df dg d 1 du
( f g ) g . f and ln( ) u .
dx dx dx dx u dx
The commonly used property for logarithmic differentiation is ln u x x ln( ).
u
To use logarithmic differentiation we must assume the function with which we take the
natural logarithm cannot be less or equal to zero ln( ) implies that f f 0.
Functions which output negative values can be solved by taking the absolute value
of the function ln f .

To apply logarithmic differentiation we simply take the logarithm on both sides of an
equation, simplify, and differentiate implicitly with respect to the independant variable.
f(x) = …

x
ln f ( ) = ln (…)
1 df d
= ln( )
x
f ( ) dx dx
df d
= f ln( )
x
dx ( ) dx
10.4 Keyword

g x
Logarithmic Differentiation: To differentiate a function of the form f x or a f x , we use
a method called Logarithmic differentiation.

10.5 Self Assessment

1. If y = x find y
x
(a) x log ex (b) e log x
x
x
(c) x log ex (d) x log ex x

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