1. **State the problem:** Differentiate the function $$y = (\ln(3x))^{17}$$ with respect to $$x$$. 2. **Formula and rules:** We will use the chain rule for differentiation, which states: $$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$ Also, the derivative of $$\ln(x)$$ is $$\frac{1}{x}$$. 3. **Apply the chain rule:** Let $$u = \ln(3x)$$, so $$y = u^{17}$$. 4. Differentiate $$y$$ with respect to $$u$$: $$\frac{dy}{du} = 17u^{16}$$ 5. Differentiate $$u$$ with respect to $$x$$: $$\frac{du}{dx} = \frac{d}{dx}[\ln(3x)] = \frac{1}{3x} \cdot 3 = \frac{1}{x}$$ 6. Combine using the chain rule: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 17(\ln(3x))^{16} \cdot \frac{1}{x}$$ 7. **Final answer:** $$y' = \frac{17(\ln(3x))^{16}}{x}$$