#### Chapter 7 Differential Calculus

Section 7.1 Derivative of a Function

# 7.1.1 Introduction

A family is going on holiday by car. The car is moving through roadworks with a velocity of $60 \mathrm{km}\left[\right]/\mathrm{h}$. The sign at the end of the roadworks says that the speed limit is, as of now, $120 \mathrm{km}\left[\right]/\mathrm{h}$. Even though the car driver puts the pedal to the metal, the velocity of the car will not jump up immediately but increase as a function of time. If the velocity increases from $60 \mathrm{km}\left[\right]/\mathrm{h}$ to $120 \mathrm{km}\left[\right]/\mathrm{h}$ in 5 seconds at a constant rate of change, then the acceleration (= change of velocity per time) equals this constant (in this case) rate of velocity change: the acceleration is the quotient of the velocity change and the time required for this change. Thus, its value is here $12$ kilometre per hour per second. In reality, the velocity of the car will not increase at a constant rate but at a time-dependent rate. If the velocity $v$ is described as a function of time $t$, then the acceleration is the slope of this function. This does not depend on the fact whether this slope is constant (in time) or not. On other words: The acceleration is the derivative of the velocity function $v$ with respect to the time $t$.
Similar relations can also be found in other technical fields such as, for example, the calculation of internal forces acting in steel frames of buildings, the forecast of atmospheric and oceanic currents, or in the modelling of financial markets, which is currently highly relevant.
This chapter reviews the basic ideas underlying these calculations, i.e. it deals with differential calculus. In other words: we will take derivatives of functions to find their slopes or rates of change. Even thought these calculations will be carried out here in a strictly mathematical way, their motivation is not purely mathematical. Derivatives, interpreted as rates of change of different functions, play an important role in many scientific fields and are often investigated as special quantities.