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*(*x*_{1}, *x*_{2}, …, *x _{n}*) of

provided the limit exists. In other words, it is obtained by differentiating *f* with respect to *x*_{1}, while treating the other variables *x*_{2}, *x*_{3}, …, *x _{n}* 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

and ...

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