# Question 44491

Feb 23, 2017

This is a product of three functions.

#### Explanation:

I use the product rule in the order:

d/dx(FS) = F'S+FS"#

(The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.)

For three functions, we have

$f \left(x\right) = u v w = u \left(v w\right)$

$f ' \left(x\right) = u ' \left(v w\right) + u \left(v w\right) '$

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

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

For four functions

$\frac{d}{\mathrm{dx}} \left(t u v w\right) = t ' u v w + t u ' v w + t u v ' w + t u v w '$

In this question $f \left(\theta\right) = \theta \cos \theta \sin \theta$

so,

$f ' \left(\theta\right) = 1 \cos \theta \sin \theta + \theta \left(- \sin \theta \sin \theta + \cos \theta \cos \theta\right)$

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

The quantity in parentheses is equal to $\cos 2 \theta$, so we can write

$f ' \left(\theta\right) = \cos \theta \sin \theta + \theta \cos 2 \theta$