Answer
$f'(\theta)=\dfrac{\sin{4\theta}}{2}.$
Work Step by Step
$u=\sin{2\theta}$; $\dfrac{du}{d\theta}=2\cos{2\theta}.$
$\dfrac{d}{du}f(u)=\dfrac{u}{2}.$
$\dfrac{d}{d\theta}f(\theta)=\dfrac{d}{du}f(u)\times\dfrac{du}{d\theta}=(2\cos{2\theta})(\dfrac{\sin{2\theta}}{2})$
$=\cos{2\theta}\sin{2\theta}=\dfrac{\sin{4\theta}}{2}.$
