# How do you differentiate f(x)=cos(3x)*(cosx) using the product rule?

Jan 7, 2016

$- \cos \left(3 x\right) \sin \left(x\right) - 3 \sin \left(3 x\right) \cos \left(x\right)$

#### Explanation:

Use the product rule $\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = f \left(x\right) g ' \left(x\right) + f ' \left(x\right) g \left(x\right)$

$f \left(x\right) = \cos \left(3 x\right)$ so $f ' \left(x\right) = - 3 \sin \left(3 x\right)$
$g \left(x\right) = \cos \left(x\right)$ so $g ' \left(x\right) = - \sin \left(x\right)$

Using the produce rule gives
f'(x) = cos(3x)*(-sin(x) +(-3sin(3x)*cos(x)
$= - \cos \left(3 x\right) \sin \left(x\right) - 3 \sin \left(3 x\right) \cos \left(x\right)$