# Derivative of Exponential Function

From ProofWiki

## Contents

## Theorem

Let $\exp$ be the exponential function.

Then:

- $D_x \left({\exp x}\right) = \exp x$

### Corollary 1

Let $c \in \R$.

Then:

- $D_x \left({\exp \left({c x}\right)}\right) = c \exp \left({c x}\right)$

### Corollary 2

Let $a \in \R: a > 0$.

Let $a^x$ be $a$ to the power of $x$.

Then:

- $D_x \left({a^x}\right) = a^x \ln a$

## Proof 1

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle D_x \left({\exp x}\right)\) | \(=\) | \(\displaystyle \) | \(\displaystyle \lim_{h \to 0} \frac {\exp \left({x+h}\right) - \exp x} h\) | \(\displaystyle \) | \(\displaystyle \) | by definition of derivative | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \lim_{h \to 0} \frac {\exp x \cdot \exp h - \exp x} h\) | \(\displaystyle \) | \(\displaystyle \) | Exponent of Sum | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \lim_{h \to 0} \frac {\exp x \left({\exp h - 1}\right)} h\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \exp x \left({\lim_{h \to 0} \frac {\exp h - 1} h}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Multiple Rule for Limits of Functions, as $\exp x$ is constant | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\displaystyle \exp x\) | \(\displaystyle \) | \(\displaystyle \) | Derivative of Exponential at Zero |

$\blacksquare$

## Proof 2

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

- $y = e^x \iff x = \ln y$

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \dfrac {\mathrm d x} {\mathrm d y}\) | \(=\) | \(\displaystyle \) | \(\displaystyle \dfrac 1 y\) | \(\displaystyle \) | \(\displaystyle \) | Derivative of Natural Logarithm Function | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \dfrac {\mathrm d y} {\mathrm d x}\) | \(=\) | \(\displaystyle \) | \(\displaystyle \dfrac {1} {1 / y}\) | \(\displaystyle \) | \(\displaystyle \) | Derivative of Inverse Function | ||

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

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

$\blacksquare$

## Proof 3

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle D_x (\ln e^x)\) | \(=\) | \(\displaystyle \) | \(\displaystyle D_x (x)\) | \(\displaystyle \) | \(\displaystyle \) | Exponential of Natural Logarithm | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \frac{1}{e^x}D_x (e^x)\) | \(=\) | \(\displaystyle \) | \(\displaystyle 1\) | \(\displaystyle \) | \(\displaystyle \) | Chain rule, Derivatives of Natural Log and Identity functions. | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle D_x (e^x)\) | \(=\) | \(\displaystyle \) | \(\displaystyle e^x\) | \(\displaystyle \) | \(\displaystyle \) | multiply both sides by $e^x$ |

$\blacksquare$

## Also see

- Equivalence of Definitions of Exponential Function where it is shown that $D_x \exp x = \exp x$ can be used to
*define*the exponential function.

## Sources