Introduction

6.1 Introduction

In Chapter 5 we developed the tools to differentiate composite functions, inverse trigonometric functions, implicit functions, and exponential and logarithmic functions. This chapter puts those tools to work: we study how the derivative is applied across engineering, the sciences, social science, and many other fields.

Over the sections ahead, the derivative will help us:

• determine the rate of change of one quantity with respect to another,
• find the equations of the tangent and normal to a curve at a given point,
• locate the turning points on a function's graph — the points that tell us where a function reaches its largest or smallest value locally,
• decide the intervals over which a function is increasing or decreasing, and
• obtain approximate values of certain quantities.

The chapter itself develops rate of change, increasing/decreasing behaviour, and maxima and minima in full as worked sections ahead; tangent-and-normal equations and derivative-based approximation are introduced here as further classic uses of the derivative.