# How do you find the derivative of y=cos(x^2) ?

Aug 6, 2014

We will need to employ the chain rule.

The chain rule states:

$\frac{d}{\mathrm{dx}} \left[f \left(g \left(x\right)\right)\right] = \frac{d}{d \left[g \left(x\right)\right]} \left[f \left(x\right)\right] \cdot \frac{d}{\mathrm{dx}} \left[g \left(x\right)\right]$

In other words, just treat ${x}^{2}$ like a whole variable, differentiate the outside function first, then multiply by the derivative of ${x}^{2}$.

We know that the derivative of $\cos u$ is $- \sin u$, where $u$ is anything - in this case it is ${x}^{2}$. And the derivative of ${x}^{2}$ is $2 x$.

(if those identities look unfamiliar to you, I may direct you to this page or this page, which have videos for the derivative of $\cos u$ and the power rule, respectively)

Anyhow, by the power rule, we now have:

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

Simplify a bit:

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