# 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 | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle e^x\) |

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 14.4$