# How do you differentiate sin(x^2)(cos(x^2))?

Sep 7, 2015

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

#### Explanation:

This problem can also be solved by directly applying the differentiation rules:

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

First, use the product rule:
$= \sin \left({x}^{2}\right) \frac{d}{\mathrm{dx}} \cos \left({x}^{2}\right) + \cos \left({x}^{2}\right) \frac{d}{\mathrm{dx}} \sin \left({x}^{2}\right)$

Then, each derivative can be solved using the chain rule:
$= - \sin \left({x}^{2}\right) \sin \left({x}^{2}\right) \frac{d}{\mathrm{dx}} {x}^{2} + \cos \left({x}^{2}\right) \cos \left({x}^{2}\right) \frac{d}{\mathrm{dx}} {x}^{2}$

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

At this point, we can simplify the expression by factoring out $2 x$ and applying the double angle identity $\cos 2 \theta = {\cos}^{2} \theta - {\sin}^{2} \theta$ :

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

$= 2 x \cos \left(2 {x}^{2}\right)$