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

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 \(\boxed{0}\).