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

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

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 $u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$. In this case, $u(\theta )={\theta }^4$ and $v(\theta )=\sin \theta$.

Step 5 ：So, we need to find $u^{\prime }(\theta )$ and $v^{\prime }(\theta )$, 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 $\boxed{{\displaystyle -{\theta }^4\cos \theta -4{\theta }^3\sin \theta }}$.