# What is the derivative of cos( sin( x ))?

Oct 20, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\sin \left(x\right)\right) \cos \left(x\right)$

#### Explanation:

$\frac{d}{\mathrm{dx}} \cos \left(\sin \left(x\right)\right)$

Using chain rule, $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$, let $u = \sin \left(x\right)$

$\frac{\mathrm{dy}}{\mathrm{du}} = \frac{d}{\mathrm{du}} \cos u = - \sin u = - \sin \left(\sin \left(x\right)\right)$

$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \sin \left(x\right) = \cos \left(x\right)$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\sin \left(x\right)\right) \cdot \cos \left(x\right) = - \sin \left(\sin \left(x\right)\right) \cos \left(x\right)$