# Derivative of Exponential Function/Proof 3

## Theorem

Let $\exp$ be the exponential function.

Then:

$\map {D_x} {\exp x} = \exp x$

## Proof

 $\displaystyle D_x (\ln e^x)$ $=$ $\displaystyle D_x (x)$ Exponential of Natural Logarithm $\displaystyle \implies \ \$ $\displaystyle \frac{1}{e^x}D_x (e^x)$ $=$ $\displaystyle 1$ Chain rule, Derivatives of Natural Log and Identity functions. $\displaystyle \implies \ \$ $\displaystyle D_x (e^x)$ $=$ $\displaystyle e^x$ multiply both sides by $e^x$

$\blacksquare$