Originally Posted by

**JeffM** We start by defining the number e. (e is not a function, but a number.) The simplest definition for our purposes is:

$e\ is\ the\ number\ such\ that\ \displaystyle \lim_{z \rightarrow 0}\dfrac{e^z - 1}{1} = 1.$ Proving that such a number exists is not obvious, but take it on faith.

Now consider the function $y = e^x.$

$e^{(x + h)} = e^x * e^h \implies$

$e^{(x + h)} - e^x= e^x * e^h - e^x = e^x(e^h - 1) \implies$

$\dfrac{e^{(x + h)} - e^x}{h} = \dfrac{e^x(e^h - 1)}{h} = e^x * \dfrac{e^h - 1}{h} \implies$

$\displaystyle \lim_{h \rightarrow 0}\dfrac{e^{(x + h)} - e^x}{h} = \lim_{h \rightarrow 0}\left(e^x * \dfrac{e^h - 1}{h}\right) = \lim_{h \rightarrow 0}\left(e^x\right) * \lim_{h \rightarrow 0}\left(\dfrac{e^h - 1}{h}\right) = e^x * 1 = e^x.$

In other words $j(x) = j'(x) \iff j(x) = e^x.$

Now try differentiating.