Chapter 5 Differentiation
The historical origin of the derivative can be traced to the problem of defining the “instantaneous velocity” of a moving particle, or, from a geometric point of view, the slope of the line tangent to a graph. But even without these practical applications, if we are given a function \(f\text{,}\) it is a natural question to ask how the output values \(f(x)\) changes when the input value \(x\) changes. More precisely, what is the ratio\((f(x)-f(c))/(x-c)\text{?}\) And if we are interested in knowing this “rate of change” at a specific input value \(c\text{,}\) the obvious thing to do is to take the limit of such ratio as \(x\) gets close to \(c\text{.}\) If \(f\) describes the position of a particle as a function of time, the limit is naturally interpreted as the instantaneous velocity of the particle. From the graphical point of view, it is the slope of line tangent to the graph of \(f\) at \(x=c\text{.}\) In this chapter, we will define the derivative and study its properties and many connections with the original function.
