# What is the significance of Euler's number?

Oct 28, 2014

Euler's number $e$ makes our lives a little easier since the slope of $f \left(x\right) = {e}^{x}$ at $x = 0$ is exactly one. In other words, $e$ is a base such that

$f ' \left(0\right) = {\lim}_{h \to 0} \frac{{e}^{0 + h} - {e}^{0}}{h} = {\lim}_{h \to 0} \frac{{e}^{h} - 1}{h} = 1$,

which is quite useful in finding $f ' \left(x\right)$. Let us use the definition to find the derivative of $f \left(x\right)$.

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{{e}^{x + h} - {e}^{x}}{h} = {\lim}_{h \to 0} \frac{{e}^{x} \cdot {e}^{h} - {e}^{x}}{h}$

by pulling ${e}^{x}$ out of the limit,

$= {e}^{x} {\lim}_{h \to 0} \frac{{e}^{h} - 1}{h} = {e}^{x} \cdot 1 = {e}^{x}$.

Hence, the derivative of ${e}^{x}$ is itself, which is very convenient.

I hope that this was helpful.