The Chemistry Maths Book, Second Edition

4.8 Logarithmic differentiation 111

Solving for gives

This is the result obtained in Example 4.6 by differentiation of the inverse function.

0 Exercises 65 – 68

4.8 Logarithmic differentiation

For some functions it is easier to differentiate the natural logarithm than the function

itself. For example, if

y 1 = 1 u

a

v

b

w

c

1- (4.17)

where u, v, w, =are functions of x, and a, b, c, =are constants, then

ln 1 y 1 = 1 a 1 ln 1 u 1 + 1 b 1 ln 1 v 1 + 1 c 1 ln 1 w 1 +1- (4.18)

and

Then, because ,

(4.19)

This method of differentiating is called logarithmic differentiation. When,y 1 = 1 uvthe

method reproduces the product rule; wheny 1 = 1 u 2 v, it reproduces the quotient rule.

EXAMPLES 4.20Logarithmic differentiation

(i)

Then

d

dx

y

y

dy

dx x x

x

ln ==

• 
  

• 
  

−













=

−

11

2

1

1

1

1

1

1

2

y

x

x

= yxx

• 
  

−













,= +−−









12

1

2

1

1

ln ln( ) ln( ) 11





1

y

dy

dx

a

u

du

dx

bd

dx

cd

d

=++ +

v

v

w

w

x



dy

dx y

dy

dx

ln

=

1

d

dx

ya

d

dx

ub

d

dx

c

d

dx

ln =++ +ln lnvwln 

dy

dx

y

=

−

1

52

4

dy

dx