Introduction to differential calculus

Mathematics Learning Centre

Introduction to Differential Calculus Christopher Thomas

c 1997

University of Sydney

Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review of Differentiation Techniques published by the Mathematics Learning Centre. I should like to thank Mary Barnes, Jackie Nicholas and Collin Phillips for their helpful comments.Christopher Thomas December 1996

Contents
1 Introduction 1.1 An example of a rate of change: velocity . . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 1.2 Constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 3 4 6 6 7 9 9

Other rates of change . . . . . . . . . . . . . . . . . . . . . . . . . .. . .

2 What is the derivative? 2.1 2.2 Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The derivative: the slope of a tangent to a graph . . . . . . . . . . . . . .

3 How do we find derivatives (in practice)? 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Derivatives of constant functions and powers . . . . . . . . . . . . . . . . .

Adding, subtracting, and multiplying bya constant . . . . . . . . . . . . . 12 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 The composite function rule (also known as the chain rule) . . . . . . . . . 15 Derivatives of exponential and logarithmic functions . . . . . . . . . . . . . 18 Derivatives of trigonometricfunctions . . . . . . . . . . . . . . . . . . . . . 21 24

4 What is differential calculus used for? 4.1 4.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Optimisation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.1 4.2.2 4.2.3 Stationary points - the idea behind optimisation . . . . . . . . . . . 24 Types of stationary points .. . . . . . . . . . . . . . . . . . . . . . 25 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 The clever idea behind differential calculus (also known as differentiation from first principles) 31 6 Solutions to exercises 35

Mathematics Learning Centre, University of Sydney

1

1

Introduction

In day to day life we are often interested in the extent towhich a change in one quantity affects a change in another related quantity. This is called a rate of change. For example, if you own a motor car you might be interested in how much a change in the amount of fuel used affects how far you have travelled. This rate of change is called fuel consumption. If your car has high fuel consumption then a large change in the amount of fuel in your tank isaccompanied by a small change in the distance you have travelled. Sprinters are interested in how a change in time is related to a change in their position. This rate of change is called velocity. Other rates of change may not have special names like fuel consumption or velocity, but are nonetheless important. For example, an agronomist might be interested in the extent to which a change in the amountof fertiliser used on a particular crop affects the yield of the crop. Economists want to know how a change in the price of a product affects the demand for that product. Differential calculus is about describing in a precise fashion the ways in which related quantities change. To proceed with this booklet you will need to be familiar with the concept of the slope (also called the gradient) of astraight line. You may need to revise this concept before continuing.

1.1
1.1.1

An example of a rate of change: velocity
Constant velocity

Figure 1 shows the graph of part of a motorist’s journey along a straight road. The vertical axis represents the distance of the motorist from some fixed reference point on the road, which could for example be the motorist’s home. Time is represented...