# Derivative of Exponential Function/Proof 2

## Theorem

Let $\exp$ be the exponential function.

Then:

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

## Proof

We use the fact that the exponential function is the inverse of the natural logarithm function:

$y = e^x \iff x = \ln y$
 $\displaystyle \dfrac {\d x} {\d y}$ $=$ $\displaystyle \dfrac 1 y$ Derivative of Natural Logarithm Function $\displaystyle \leadsto \ \$ $\displaystyle \dfrac {\d y} {\d x}$ $=$ $\displaystyle \dfrac 1 {1 / y}$ Derivative of Inverse Function $\displaystyle$ $=$ $\displaystyle y$ $\displaystyle$ $=$ $\displaystyle e^x$

$\blacksquare$