6 - Partial derivatives

Summary

The final draft of a mathematics book contained the sentence: ‘∂f/∂x means the ratio at constant y of δf and δx, where df and δx are vanishingly small’ When the author received the publisher's proofs, this appeared as ‘∂f/∂x means the ratio at constant y of . and ‥’ On closer examination with a powerful magnifying glass, however, it turned out that the first two full stops were in fact the smallest δf and δx that the publisher was able to produce.

Many branches of science involve partial derivatives. The aim of this chapter is to make you understand what they are, and become so fluent at manipulating them that this sort of operation becomes as familiar and as accepted as the arithmetic operations with ordinary numbers. This will then enable you to concentrate on the basic principles of your science problem, rather than battling with the mathematics of the partial derivatives involved.

Introduction

We are often interested in calculating the derivatives df/dx, d²f/dx², etc for a function f(x) of a single variable x. Similarly, for functions of more than one variable f(x, y, …), we may well also want the derivatives. These are written as, for example, ∂f/∂x, which means ‘the rate of change of the function f with respect to small changes in x, assuming that all the other independent variables are kept constant’.