# Derivative of Exponential Function/Proof 3

## Theorem

Let $\exp$ be the exponential function.

Then:

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

## Proof

 $\displaystyle \map {D_x} {\ln e^x}$ $=$ $\displaystyle \map {D_x} x$ Exponential of Natural Logarithm $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {e^x} \map {D_x} {e^x}$ $=$ $\displaystyle 1$ Chain Rule for Derivatives, Derivative of Natural Logarithm Function, Derivative of Identity Function $\displaystyle \leadsto \ \$ $\displaystyle \map {D_x} {e^x}$ $=$ $\displaystyle e^x$ multiply both sides by $e^x$

$\blacksquare$