Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

and using (2.6), we obtain, as before omitting the argument,

df

dx

=u

d

dx

(vw)+

du

dx

vw.

Using (2.6) again to expand the first term on the RHS gives the complete result

d

dx

(uvw)=uv

dw

dx

+u

dv

dx

w+

du

dx

vw (2.9)

or

(uvw)′=uvw′+uv′w+u′vw. (2.10)

It is readily apparent that this can be extended to products containing any number

nof factors; the expression for the derivative will then consist ofnterms with

the prime appearing in successive terms on each of thenfactors in turn. This is

probably the easiest way to recall the product rule.

2.1.3 The chain rule

Products are just one type of complicated function that we may encounter in

differentiation. Another is the function of a function, e.g.f(x)=(3+x^2 )^3 =u(x)^3 ,

whereu(x)=3+x^2 .If∆f,∆uand ∆xare small finite quantities, it follows that

∆f

∆x

=

∆f

∆u

∆u

∆x

;

As the quantities become infinitesimally small we obtain

df

dx

=

df

du

du

dx

. (2.11)

This is thechain rule, which we must apply when differentiating a function of a

function.

Find the derivative with respect toxoff(x)=(3+x^2 )^3.

Rewriting the function asf(x)=u^3 ,whereu(x)=3+x^2 , and applying (2.11) we find

df

dx

=3u^2

du

dx

=3u^2

d

dx

(3 +x^2 )=3u^2 × 2 x=6x(3 +x^2 )^2 .

Similarly, the derivative with respect toxoff(x)=1/v(x) may be obtained by

rewriting the function asf(x)=v−^1 and applying (2.11):

df

dx

=−v−^2

dv

dx

=−

1

v^2

dv

dx

. (2.12)

The chain rule is also useful for calculating the derivative of a functionfwith

respect toxwhen bothxandfare written in terms of a variable (or parameter),

sayt.