The Derivative

Abstract

The geometric motivation for the derivative is most likely familiar territory. Given a function g(x), the derivative g′(x) is understood to be the slope of the graph of g at each point x in the domain. A graphical picture (Fig. 5.1) reveals the impetus behind the mathematical definition

$$g'\left( c \right) = \mathop {\lim }\limits_{x \to c} \frac{{g\left( x \right) - g\left( c \right)}}{{x - c}}.$$