Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

can be obtained. It will be noticed that the first bracket in (5.3) actually approxi-

mates to∂f(x, y+∆y)/∂xbut that this has been replaced by∂f(x, y)/∂xin (5.4).

This approximation clearly has the same degree of validity as that which replaces

the bracket by the partial derivative.

How valid an approximation (5.4) is to (5.3) depends not only on how small

∆xand ∆yare but also on the magnitudes of higher partial derivatives; this is

discussed further in section 5.7 in the context of Taylor series for functions of

more than one variable. Nevertheless, letting the small changes ∆xand ∆yin

(5.4) become infinitesimal, we can define thetotal differentialdfof the function

f(x, y), without any approximation, as

df=

∂f

∂x

dx+

∂f

∂y

dy. (5.5)

Equation (5.5) can be extended to the case of a function ofn variables,

f(x 1 ,x 2 ,...,xn);

df=

∂f

∂x 1

dx 1 +

∂f

∂x 2

dx 2 +···+

∂f

∂xn

dxn. (5.6)

Find the total differential of the functionf(x, y)=yexp(x+y).

Evaluating the first partial derivatives, we find

∂f

∂x

=yexp(x+y),

∂f

∂y

=exp(x+y)+yexp(x+y).

Applying (5.5), we then find that the total differential is given by

df=[yexp(x+y)]dx+[(1+y)exp(x+y)]dy.

In some situations, despite the fact that several variablesxi,i=1, 2 ,...,n,

appear to be involved, effectively only one of them is. This occurs if there are

subsidiary relationships constraining all thexito have values dependent on the

value of one of them, sayx 1. These relationships may be represented by equations

that are typically of the form

xi=xi(x 1 ),i=2, 3 ,...,n. (5.7)

In principlefcan then be expressed as a function ofx 1 alone by substituting

from (5.7) forx 2 ,x 3 ,...,xn, and then thetotal derivative(or simply the derivative)

offwith respect tox 1 is obtained by ordinary differentiation.

Alternatively, (5.6) can be used to give

df

dx 1

=

∂f

∂x 1

+

(

∂f

∂x 2

)

dx 2

dx 1

+···+

(

∂f

∂xn

)

dxn

dx 1

. (5.8)

It should be noted that the LHS of this equation is the total derivativedf/dx 1 ,

whilst the partial derivative∂f/∂x 1 forms only a part of the RHS. In evaluating