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 $f(x)$. $$\displaystyle \dfrac{d}{dx}e^x = e^x$$ $$\displaystyle \dfrac{d}{dx}e^{f(x)} = e^{f(x)} \times f'(x)$$

Example 1

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

Example 2

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

Example 3

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

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 $\dfrac{d}{dx}\sin{x} = \cos{x}$.

Example 5

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

Example 6

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