1. **State the problem:** Find the derivative of the function $f(x) = x^3 \cdot \ln(x)$. 2. **Recall the product rule:** For two functions $u(x)$ and $v(x)$, the derivative of their product is given by $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Identify parts:** Here, $u(x) = x^3$ and $v(x) = \ln(x)$. 4. **Compute derivatives:** $$u'(x) = 3x^2$$ $$v'(x) = \frac{1}{x}$$ 5. **Apply product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = 3x^2 \cdot \ln(x) + x^3 \cdot \frac{1}{x}$$ 6. **Simplify terms:** $$x^3 \cdot \frac{1}{x} = x^{3-1} = x^2$$ 7. **Combine:** $$f'(x) = 3x^2 \ln(x) + x^2$$ 8. **Factor out common term $x^2$:** $$f'(x) = x^2 (3 \ln(x) + 1)$$ **Final answer:** $$\boxed{f'(x) = x^2 (3 \ln(x) + 1)}$$