# How do you differentiate f(x)=4/(x+1)^2  using the chain rule?

Oct 30, 2016

$f ' \left(x\right) = - \frac{8}{x + 1} ^ 3$

#### Explanation:

Express $f \left(x\right) = 4 {\left(x + 1\right)}^{-} 2$

differentiate using the $\textcolor{b l u e}{\text{chain rule}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

let $u = x + 1 \Rightarrow \frac{\mathrm{du}}{\mathrm{dx}} = 1$

and $y = 4 {u}^{-} 2 \Rightarrow \frac{\mathrm{dy}}{\mathrm{du}} = - 8 {u}^{-} 3$

substitute these values into $\frac{\mathrm{dy}}{\mathrm{dx}}$ writing u in terms of x.

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 8 {u}^{-} 3 \times 1 = - \frac{8}{x + 1} ^ 3$