Introducing e

The most useful base for the calculus is the number e. e is defined as the real number such that the derivative of e^x is e^x itself,

In other words, e is the real number such that the constant

(where Δx is a nonzero infinitesimal). It will be shown in Section 8.3 that there is such a number e and that e has the approximate value

e ~ 2.71828.

The function y = e^x is called the exponential function. e^x is always positive and follows the rules

e^a+b = e^a · e^b,   e^a·b = (e^a)^b,   e⁰ = 1.

Figure 2.5.7 shows the graph of y = e^x.