# Derivative of Exponential Function

## 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 $e^x$

$\blacksquare$