In this chapter we generalise the discussion of differential calculus in Chapter 3 to functions of more than one variable. Many results will be taken over from Chapter 3 and will be dealt with rather briefly, so that we can focus on the differences between the two cases.
Given a function f(x1, x2, …, xn) of n independent variables, x1, x2, …, xn, the partial derivative of f with respect to x1 is defined by
provided the limit exists. In other words, it is obtained by differentiating f with respect to x1, while treating the other variables x2, x3, …, xn as fixed parameters. Partial derivatives with respect to the other variables are defined in a similar way. For example, if
then differentiating with respect to x keeping y fixed gives, using the product rule (3.20),
while differentiating with respect to y keeping x fixed gives
Higher derivatives are obtained by repeated partial differentiation, so that
for the second derivatives. Thus for the function (7.2), using (7.3) one obtains