### Introduction

When a function of more than one independent input variable changes because of changes in one or more of the input variables, it is important to calculate the change in the function itself. This can be investigated by holding all but one of the variables constant and finding the rate of change of the function with respect to the one remaining variable. This process is called partial differentiation. In this Section we show how to carry out the process.

#### Prerequisites

- understand the principle of differentiating a function of one variable

#### Learning Outcomes

- understand the concept of partial differentiation
- differentiate a function partially with respect to each of its variables in turn
- evaluate first partial derivatives
- carry out successive partial differentiations
- formulate second partial derivatives

#### Contents

1 First partial derivatives1.1 The $x$ partial derivative

1.2 The $y$ partial derivative

1.3 Functions of several variables

2 Second partial derivatives

2.1 Mixed second derivatives

3 Engineering Example 1

3.1 The ideal gas law and Redlich-Kwong equation