Chapter 31
Logarithmic Differentiation
31.1 Introduction to logarithmic
differentiation
With certain functions containing more complicated
products and quotients, differentiation is often made
easier if the logarithm of the function is taken before
differentiating. This technique, called ‘logarithmic
differentiation’ is achieved with a knowledge of (i) the
laws of logarithms, (ii) the differential coefficients of
logarithmic functions, and (iii) the differentiation of
implicit functions.
31.2 Laws of logarithms
Three laws of logarithms may be expressed as:
(i) log(A × B) = log A + log B
(ii) log
A
B
= log A − log B
(iii) log A n = n log A
In calculus, Napierian logarithms (i.e. logarithms to a
base of ‘e’) are invariably used. Thus for two functions f (x) and g(x) the laws of logarithms may be
expressed as:
(i) ln[ f (x) · g(x)] = ln f (x) + ln g(x)
(ii) ln
f (x)
g(x)
= ln f (x) − ln g(x)
(iii) ln[ f (x)] n = n ln f (x)
Taking Napierian logarithms of both sides of the equation y =
f (x) · g(x)
h(x)
gives:
ln y = ln
f (x) · g(x)
h(x)
which may be simplified using the above laws of
logarithms, giving:
ln y = ln f (x) + ln g(x) − ln h(x)
This latter form of the equation is often easier to
differentiate.
31.3 Differentiation of logarithmic
functions
The differential coefficient of the logarithmic function
ln x is given by:
d
dx
(lnx) =
1
x
More generally, it may be shown that:
d
dx
[ln f (x)] =
f
(x)
f (x)
(1)
For example, if y = ln(3x 2 + 2x − 1) then,
dy
dx
=
6x + 2
3x 2 + 2x − 1
Similarly, if y = ln(sin 3x) then
dy
dx
=
3 cos 3x
sin 3x
= 3 cot 3x.
Now try the following exercise
Exercise 131 Further problems on
differentiating logarithmic functions
Differentiate the following using the laws for
logarithms.
1. ln (4x − 10)
4
4x − 10
Logarithmic Differentiation
31.1 Introduction to logarithmic
differentiation
With certain functions containing more complicated
products and quotients, differentiation is often made
easier if the logarithm of the function is taken before
differentiating. This technique, called ‘logarithmic
differentiation’ is achieved with a knowledge of (i) the
laws of logarithms, (ii) the differential coefficients of
logarithmic functions, and (iii) the differentiation of
implicit functions.
31.2 Laws of logarithms
Three laws of logarithms may be expressed as:
(i) log(A × B) = log A + log B
(ii) log
A
B
= log A − log B
(iii) log A n = n log A
In calculus, Napierian logarithms (i.e. logarithms to a
base of ‘e’) are invariably used. Thus for two functions f (x) and g(x) the laws of logarithms may be
expressed as:
(i) ln[ f (x) · g(x)] = ln f (x) + ln g(x)
(ii) ln
f (x)
g(x)
= ln f (x) − ln g(x)
(iii) ln[ f (x)] n = n ln f (x)
Taking Napierian logarithms of both sides of the equation y =
f (x) · g(x)
h(x)
gives:
ln y = ln
f (x) · g(x)
h(x)
which may be simplified using the above laws of
logarithms, giving:
ln y = ln f (x) + ln g(x) − ln h(x)
This latter form of the equation is often easier to
differentiate.
31.3 Differentiation of logarithmic
functions
The differential coefficient of the logarithmic function
ln x is given by:
d
dx
(lnx) =
1
x
More generally, it may be shown that:
d
dx
[ln f (x)] =
f
(x)
f (x)
(1)
For example, if y = ln(3x 2 + 2x − 1) then,
dy
dx
=
6x + 2
3x 2 + 2x − 1
Similarly, if y = ln(sin 3x) then
dy
dx
=
3 cos 3x
sin 3x
= 3 cot 3x.
Now try the following exercise
Exercise 131 Further problems on
differentiating logarithmic functions
Differentiate the following using the laws for
logarithms.
1. ln (4x − 10)
4
4x − 10