# How do you differentiate y=e^(-x/4)?

Mar 4, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {e}^{- \frac{x}{4}} / 4$

#### Explanation:

In order to differentiate a function of a function, we use the chain rule. In other words, if $y = f \left(u\right)$ and $u = f \left(x\right)$, then

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

In this case, let $u = - \frac{x}{4}$ and $y = {e}^{u}$. Then

$\frac{\mathrm{dy}}{\mathrm{du}} = {e}^{u} = {e}^{- \frac{x}{4}}$;

$\frac{\mathrm{du}}{\mathrm{dx}} = - \frac{1}{4}$

and, importantly,

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}} = {e}^{- \frac{x}{4}} \left(- \frac{1}{4}\right) = - {e}^{- \frac{x}{4}} / 4$