5

Partial differentiation

In chapter 2, we discussed functionsfof only one variablex, which were usually

writtenf(x). Certain constants and parameters may also have appeared in the

definition off,e.g.f(x)=ax+ 2 contains the constant 2 and the parametera, but

onlyxwas considered as a variable and only the derivativesf(n)(x)=dnf/dxn

were defined.

However, we may equally well consider functions that depend on more than one

variable, e.g. the functionf(x, y)=x^2 +3xy, which depends on the two variables

xandy. For any pair of valuesx, y, the functionf(x, y) has a well-defined value,

e.g.f(2,3) = 22. This notion can clearly be extended to functions dependent on

more than two variables. For then-variable case, we writef(x 1 ,x 2 ,...,xn)for

a function that depends on the variablesx 1 ,x 2 ,...,xn.Whenn=2,x 1 andx 2

correspond to the variablesxandyused above.

Functions of one variable, likef(x), can be represented by a graph on a

plane sheet of paper, and it is apparent that functions of two variables can,

with little effort, be represented by a surface in three-dimensional space. Thus,

we may also picturef(x, y) as describing the variation of height with position

in a mountainous landscape. Functions of many variables, however, are usually

very difficult to visualise and so the preliminary discussion in this chapter will

concentrate on functions of just two variables.

5.1 Definition of the partial derivative

It is clear that a functionf(x, y) of two variables will have a gradient in all

directions in thexy-plane. A general expression for this rate of change can be

found and will be discussed in the next section. However, we first consider the

simpler case of finding the rate of change off(x, y) in the positivex-andy-

directions. These rates of change are called thepartial derivativeswith respect