Answer
$$ R'(\theta)=\frac{-4\sin \theta}{(4+\cos \theta)^2}.$$
Work Step by Step
Since $ R(\theta)=\frac{\cos\theta}{4+\cos \theta}$, then using the quotient rule, we have
$$ R'(\theta)=\frac{(4+\cos \theta)(-\sin \theta)-\cos \theta (-\sin \theta)}{(4+\cos \theta)^2}=\frac{-4\sin \theta}{(4+\cos \theta)^2}.$$
