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

Aug 4, 2014

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 2 x \sin {x}^{2}$

Process:

This problem will require use of the chain rule.

If $y = \cos {x}^{2}$, then, by the chain rule, the derivative will be equal to the derivative of $\cos {x}^{2}$ with respect to ${x}^{2}$, multiplied by the derivative of ${x}^{2}$ with respect to $x$.

We know the basic identity $\frac{d}{\mathrm{dx}} \left[\cos x\right] = - \sin x$. And, the power rule gives us $\frac{d}{\mathrm{dx}} \left[{x}^{2}\right] = 2 x$.

(if those identities look unfamiliar to you, some excellent videos can be located here and here, which explain the identity for $\cos x$ and the power rule, respectively)

So, the derivative of $\cos {x}^{2}$ will therefore be:

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

Which further simplifies to:

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