# How do you differentiate f(x) = x*e^(x^2)*sin(x^2) using the product rule?

Oct 27, 2015

Use the product rule twice (or the three factor product rule once).

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(u v w\right) = \frac{d}{\mathrm{dx}} \left(u \left(v w\right)\right)$

$= u ' \left(v w\right) + u \left[\frac{d}{\mathrm{dx}} \left(v w\right)\right]$

$= u ' \left(v w\right) + u \left[v ' w + v w '\right]$

$= u ' v w + u v ' w + u v w '$

Apply either the formula or the process to this question with

$u = x$ $\text{ }$ $\text{ }$ $\text{ }$$\text{ }$ So $u ' = x$
$v = {e}^{{x}^{2}}$ $\text{ }$ $\text{ }$$\text{ }$$\text{ }$ So $v ' = 2 x {e}^{{x}^{2}}$
$w = \sin \left({x}^{2}\right)$ $\text{ }$ $\text{ }$ So $w ' = 2 x \cos \left({x}^{2}\right)$