# Calculus in a Nutshell: Functions and their Derivatives

When I was about thirteen, the library was going to get ‘Calculus for the Practical Man.’ By this time I knew, from reading the encyclopedia, that calculus was an important and interesting subject, and I ought to learn it.

Richard P. Feynman, from What Do You Care What Other People Think?

## Introduction

Calculus is an important branch of mathematics that deals with the methods for calculating derivatives and integrals of functions and using this information to study the properties of functions. It was independently invented by I. Newton and W. Leibniz in the 18 ${^{th}}$ century and it was further developed by other great mathematicians in the centuries that follows (see Figure below). Figure 1: Some of the great mathematician that invented the Calculus.

It comprises two areas:

• Differential calculus ${\rightarrow}$ It concerns the study of the rate of variation of functions.
• Integral calculus ${\rightarrow}$ It concern the study of the area under functions.

Depending on the nature of the functions involved in the calculations, we can further distinguish between the single- and multi-variable calculus. In this chapter, the main concepts and methods of the single-variable calculus are summarised.