1. **State the problem:** Differentiate the function $$y = (\sin x)^{x^3}$$ with respect to $$x$$. 2. **Formula and approach:** When differentiating a function of the form $$y = u(x)^{v(x)}$$, use logarithmic differentiation: $$\ln y = v(x) \ln u(x)$$ Then differentiate both sides using the product and chain rules. 3. **Apply logarithmic differentiation:** $$\ln y = x^3 \ln(\sin x)$$ 4. **Differentiate both sides:** $$\frac{1}{y} \frac{dy}{dx} = 3x^2 \ln(\sin x) + x^3 \frac{1}{\sin x} \cos x$$ Here, we used the product rule on the right side and the derivative of $$\ln(\sin x)$$ is $$\frac{\cos x}{\sin x}$$. 5. **Solve for $$\frac{dy}{dx}$$:** $$\frac{dy}{dx} = y \left(3x^2 \ln(\sin x) + x^3 \cot x \right)$$ 6. **Substitute back $$y$$:** $$\frac{dy}{dx} = (\sin x)^{x^3} \left(3x^2 \ln(\sin x) + x^3 \cot x \right)$$ **Final answer:** $$\boxed{\frac{dy}{dx} = (\sin x)^{x^3} \left(3x^2 \ln(\sin x) + x^3 \cot x \right)}$$