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 $$y = f(x)^{g(x)}$$, use logarithmic differentiation. 3. **Step 1: Take the natural logarithm of both sides:** $$\ln y = \ln \left((\sin x)^{x^3}\right) = x^3 \ln(\sin x)$$ 4. **Step 2: Differentiate both sides with respect to $$x$$:** Using the chain rule on the left and product rule on the right, $$\frac{1}{y} \frac{dy}{dx} = 3x^2 \ln(\sin x) + x^3 \frac{1}{\sin x} \cos x$$ 5. **Step 3: Solve for $$\frac{dy}{dx}$$:** Multiply both sides by $$y$$: $$\frac{dy}{dx} = y \left(3x^2 \ln(\sin x) + x^3 \cot x \right)$$ 6. **Step 4: Substitute back $$y = (\sin x)^{x^3}$$:** $$\frac{dy}{dx} = (\sin x)^{x^3} \left(3x^2 \ln(\sin x) + x^3 \cot x \right)$$ **Final answer:** $$\boxed{\frac{dy}{dx} = (\sin x)^{x^3} \left(3x^2 \ln(\sin x) + x^3 \cot x \right)}$$ This method uses logarithmic differentiation to handle the variable exponent and base, applying the product and chain rules carefully.