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

Nov 23, 2015

$- 2 \sin \left({x}^{2}\right) - 4 {x}^{2} \cos \left({x}^{2}\right)$

#### Explanation:

This will require the chain rule. Recall that $\frac{d}{\mathrm{dx}} \left[\cos \left(u\right)\right] = - u ' \sin \left(u\right)$.

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

To find the second derivative, we must use the product rule.

$y ' ' = \sin \left({x}^{2}\right) \frac{d}{\mathrm{dx}} \left[- 2 x\right] + \left(- 2 x\right) \frac{d}{\mathrm{dx}} \left[\sin \left({x}^{2}\right)\right]$
$y ' ' = - 2 \sin \left({x}^{2}\right) - 2 x \cos \left({x}^{2}\right) \cdot \frac{d}{\mathrm{dx}} \left[{x}^{2}\right]$
$y ' ' = - 2 \sin \left({x}^{2}\right) - 2 x \cos \left({x}^{2}\right) \cdot 2 x$
$y ' ' = - 2 \sin \left({x}^{2}\right) - 4 {x}^{2} \cos \left({x}^{2}\right)$