itomath.com

Proof of the Derivative of $e^x$

The derivative of $f\left(x\right)=e^x$ is $\displaystyle f'(x)=e^x$.
To prove this, we will use the following properties:
  1. $\displaystyle f'(x)=\lim \limits_{h \to 0} \frac{f\left(x+h\right)-f\left(x\right)}{h}$
  2. $e^{x}e^{y}=e$