1. **State the problem:** Find the differential $dy$ of the function $y = \csc(3x)$.\n\n2. **Recall the formula:** The derivative of $\csc(u)$ with respect to $x$ is $\frac{d}{dx}[\csc(u)] = -\csc(u)\cot(u) \cdot \frac{du}{dx}$.\n\n3. **Identify the inner function:** Here, $u = 3x$, so $\frac{du}{dx} = 3$.\n\n4. **Apply the chain rule:**\n$$\frac{dy}{dx} = -\csc(3x) \cot(3x) \cdot 3 = -3 \csc(3x) \cot(3x)$$\n\n5. **Express the differential:** Since $dy = \frac{dy}{dx} dx$, we have\n$$dy = -3 \csc(3x) \cot(3x) \, dx$$\n\n**Final answer:**\n$$dy = -3 \csc(3x) \cot(3x) \, dx$$