Answer
The Derivative is:
$\frac{dr}{d\theta}=-\theta(2\sin\theta+\theta\cos\theta)$
Work Step by Step
$r=4-\theta^2\sin\theta$
Applying Derivative rules:
$y'=f'(x)+g'(x)$ $and$ $g'(x)=h'(x)\cdot v(x)+h(x)\cdot v'(x)$
$\frac{dr}{d\theta}=\frac{d}{d\theta}(4)-\frac{d}{d\theta}(\theta^2\sin\theta)$
$\frac{dr}{d\theta}=(0)-((2\theta^{2-1})(\sin\theta)+\theta^2(\cos\theta)$
$\frac{dr}{d\theta}=-2\theta\sin\theta-\theta^2\cos\theta$
$\frac{dr}{d\theta}=-\theta(2\sin\theta+\theta\cos\theta)$
