1. **State the problem:** Find the derivative of the function $y = \sin(\cot(5x))$ with respect to $x$. 2. **Recall the chain rule:** If $y = \sin(u)$ where $u = \cot(5x)$, then by the chain rule, $$\frac{dy}{dx} = \cos(u) \cdot \frac{du}{dx}.$$ 3. **Differentiate the inner function:** We need $\frac{d}{dx} \cot(5x)$. Recall that $$\frac{d}{dx} \cot(v) = -\csc^2(v) \cdot \frac{dv}{dx}.$$ Here, $v = 5x$, so $$\frac{dv}{dx} = 5.$$ Therefore, $$\frac{d}{dx} \cot(5x) = -\csc^2(5x) \cdot 5 = -5 \csc^2(5x).$$ 4. **Combine results:** Substitute back into the chain rule expression: $$\frac{dy}{dx} = \cos(\cot(5x)) \cdot (-5 \csc^2(5x)) = -5 \cos(\cot(5x)) \csc^2(5x).$$ 5. **Final answer:** $$\boxed{\frac{dy}{dx} = -5 \cos(\cot(5x)) \csc^2(5x)}.$$