1. **State the problem:** Differentiate the function $$f(x) = \frac{\ln x}{x^3}$$. 2. **Rewrite the function:** Using the property $$\frac{1}{x^3} = x^{-3}$$, rewrite the function as $$f(x) = \ln x \cdot x^{-3}$$. 3. **Use the product rule:** The product rule states that $$\frac{d}{dx}[u \cdot v] = u'v + uv'$$. Here, let $$u = \ln x$$ and $$v = x^{-3}$$. 4. **Find derivatives of u and v:** - $$u' = \frac{1}{x}$$ - $$v' = -3x^{-4}$$ (using the power rule) 5. **Apply the product rule:** $$\frac{d}{dx} \left( \ln x \cdot x^{-3} \right) = \frac{1}{x} \cdot x^{-3} + \ln x \cdot (-3x^{-4})$$ 6. **Simplify each term:** - $$\frac{1}{x} \cdot x^{-3} = x^{-4}$$ - $$\ln x \cdot (-3x^{-4}) = -3x^{-4} \ln x$$ 7. **Combine terms:** $$f'(x) = x^{-4} - 3x^{-4} \ln x = x^{-4} (1 - 3 \ln x)$$ **Final answer:** $$\boxed{f'(x) = \frac{1 - 3 \ln x}{x^4}}$$