# What is the derivative of this function cos x^5?

Sep 11, 2016

$- 5 {x}^{4} \sin \left({x}^{5}\right)$

#### Explanation:

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

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

let $u = {x}^{5} \Rightarrow \frac{\mathrm{du}}{\mathrm{dx}} = 5 {x}^{4}$

and $y = \cos u \Rightarrow \frac{\mathrm{dy}}{\mathrm{du}} = - \sin u$

substitute these values into (A) change u back to terms of x.

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = - \sin u \times 5 {x}^{4} = - 5 {x}^{4} \sin \left({x}^{5}\right)$