# How do you find the derivative of e^(-3x)?

##### 2 Answers
Jun 18, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 3 {e}^{- 3 x}$

#### Explanation:

using the chain rule

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \textcolor{red}{\frac{\mathrm{du}}{\mathrm{dx}}}$

$y = {e}^{- 3 x}$

$\textcolor{red}{u = - 3 x \implies \frac{\mathrm{dy}}{\mathrm{du}} = - 3}$

$\frac{\mathrm{dy}}{\mathrm{du}} = \frac{d}{\mathrm{du}} \left({e}^{u}\right) = {e}^{u}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \textcolor{red}{\frac{\mathrm{du}}{\mathrm{dx}}} = {e}^{u} \times \textcolor{red}{\left(- 3\right)}$

$= - 3 {e}^{u} = - 3 {e}^{- 3 x}$

in general:

$\frac{d}{\mathrm{dx}} \left({e}^{f \left(x\right)}\right) = f ' \left(x\right) {e}^{f \left(x\right)}$

Jun 22, 2018

color(brown)(f'(x) = -3 e^-(3x)

#### Explanation:

$f \left(x\right) = {e}^{-} \left(3 x\right)$

$f ' \left(x\right) = \left(\frac{d}{\mathrm{dx}}\right) {e}^{-} \left(3 x\right)$

$f ' \left(x\right) = {e}^{- 3 x} \cdot \left(\frac{d}{\mathrm{dx}}\right) \left(- 3 x\right)$

$f ' \left(x\right) = {e}^{-} \left(3 x\right) \cdot - 3$

color(brown)(f'(x) = -3 e^-(3x)