Use implicit differentiation to find $\frac{dr}{d\theta }$
$$\cot \left(r{\theta }^4\right)=\frac{1}{7}$$

Step 1 ：First, we rewrite the given equation as $\tan (r{\theta }^4)=7$.

Step 2 ：Next, we take the derivative of both sides with respect to $\theta$.

Step 3 ：On the left side, we use the chain rule to differentiate $\tan (r{\theta }^4)$. The derivative of $\tan (x)$ is ${\sec }^2(x)$, and the derivative of $r{\theta }^4$ is $4r{\theta }^3\frac{dr}{d\theta }$.

Step 4 ：So, the derivative of the left side is ${\sec }^2(r{\theta }^4)\cdot 4r{\theta }^3\frac{dr}{d\theta }$.

Step 5 ：On the right side, the derivative of 7 with respect to $\theta$ is 0.

Step 6 ：So, we have the equation ${\sec }^2(r{\theta }^4)\cdot 4r{\theta }^3\frac{dr}{d\theta }=0$.

Step 7 ：We can simplify this equation by dividing both sides by ${\sec }^2(r{\theta }^4)\cdot 4r{\theta }^3$, which gives us $\frac{dr}{d\theta }=0$.

Step 8 ：Thus, the derivative of $r$ with respect to $\theta$ is $\boxed{{\displaystyle 0}}$.