# Derivative of Exponential Functions

The functions $e^{-x}$, $e^{3x+2}$ and $e^{x^2+2x-1}$ are all of the form $e^{f(x)}$.
$e^{f(x)} \gt 0$ for all $x$, no matter what the function $f(x)$.
\large \displaystyle \begin{align} \dfrac{d}{dx}e^x &= e^x \\ \dfrac{d}{dx}e^{f(x)} &= e^{f(x)} \times f'(x) \end{align}

### Example 1

Find $\displaystyle \dfrac{dy}{dx}$ if $y=e^{4x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= e^{4x} \times \dfrac{d}{dx}4x \\ &= e^{4x} \times 4 \\ &= 4e^{4x} \end{align}

### Example 2

Find $\displaystyle \dfrac{dy}{dx}$ if $y=x^2e^{3x}$.

\begin{align} \displaystyle \require{AMSsymbols} \require{color} \dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times e^{3x} + x^2 \times \dfrac{d}{dx}e^{3x} &\color{red} \text{product rule}\\ &= 2x \times e^{3x} + x^2 \times e^{3x} \times \dfrac{d}{dx}3x \\ &= 2xe^{3x} + x^2 \times e^{3x} \times 3 \\ &= 2xe^{3x} + 3x^2 e^{3x} \end{align}

### Example 3

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\dfrac{e^{2x}}{x}$.

\begin{align} \displaystyle \require{AMSsymbols} \require{color} \dfrac{dy}{dx} &= \dfrac{\dfrac{d}{dx}e^{2x} \times x-e^{2x} \times \dfrac{d}{dx}x}{x^2} &\color{red} \text{quotient rule} \\ &= \dfrac{2e^{2x} \times x-e^{2x} \times 1}{x^2} \\ &= \dfrac{2xe^{2x}-e^{2x}}{x^2} \end{align}

## Extension Examples

These Extension Examples require to have some prerequisite skills, including;
\begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \dfrac{d}{dx}\log_e{x} &= \dfrac{1}{x} \\ \end{align}

### Example 4

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^x\sin{x}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}e^x \times \sin{x} + e^x \times \dfrac{d}{dx}\sin{x} \\ &= e^x \times \sin{x} + e^x \times \cos{x} \\ &= e^x\sin{x} + e^x\cos{x} \end{align}

### Example 5

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^{\cos{x}}$, known that $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= e^{\cos{x}} \times \dfrac{d}{dx}\cos{x} \\ &= e^{\cos{x}} \times -\sin{x} \\ &= -e^{\cos{x}}\sin{x} \end{align}

### Example 6

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^{2x}\log_e{x^2}$, known that $\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x}$.

\begin{align} \displaystyle \dfrac{dy}{dx} &= \dfrac{d}{dx}e^{2x} \times \log_e{x^2} + e^{2x} \times \dfrac{d}{dx}\log_e{x^2} \\ &= 2e^{2x} \times \log_e{x^2} + e^{2x} \times \dfrac{1}{x^2} \times \dfrac{d}{dx}x^2 \\ &= 2e^{2x}\log_e{x^2} + e^{2x} \times \dfrac{1}{x^2} \times 2x \\ &= 2e^{2x}\log_e{x^2} + \dfrac{2e^{2x}}{x} \end{align}