Find $\frac{d r}{d \theta}$.
\[
r=4-\theta^{4} \sin \theta
\]

Step 1 ：We are given the function \(r = 4 - \theta^4 \sin \theta\) and we are asked to find \(\frac{d r}{d \theta}\).

Step 2 ：This function is a difference of two functions, \(4\) and \(\theta^4 \sin \theta\). The derivative of a constant is zero, so the derivative of \(4\) is \(0\).

Step 3 ：For the second function, \(\theta^4 \sin \theta\), we need to apply the product rule because it is the product of \(\theta^4\) and \(\sin \theta\).

Step 4 ：The product rule states that the derivative of \(u(x)v(x)\) is \(u'(x)v(x) + u(x)v'(x)\). In this case, \(u(\theta) = \theta^4\) and \(v(\theta) = \sin \theta\).

Step 5 ：So, we need to find \(u'(\theta)\) and \(v'(\theta)\), and then substitute them into the product rule formula.

Step 6 ：The derivative of \(\theta^4\) is \(4\theta^3\) and the derivative of \(\sin \theta\) is \(\cos \theta\).

Step 7 ：Substituting these into the product rule formula, we get \(- \theta^{4} \cos \theta - 4 \theta^{3} \sin \theta\).

Step 8 ：Final Answer: The derivative of \(r\) with respect to \(\theta\) is \(\boxed{- \theta^{4} \cos \theta - 4 \theta^{3} \sin \theta}\).