# What is the derivative of cos(x^2)?

Mar 11, 2016

$\frac{d}{\mathrm{dx}} \cos \left({x}^{2}\right) = - 2 x \sin \left({x}^{2}\right)$

#### Explanation:

Let's use the chain rule

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

In other words:

$\frac{d}{\mathrm{dx}} \cos \left({x}^{2}\right) = \frac{d}{\mathrm{du}} \cos \left(u\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

where

$u = {x}^{2}$

doing each part individually:

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

and

$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} {x}^{2} = 2 x$

putting this back together we get

$\frac{d}{\mathrm{dx}} \cos \left({x}^{2}\right) = - 2 x \sin \left({x}^{2}\right)$