1. Problem: Find the derivative of the function $$f(x) = \exp(x) x^2$$ for $$x > 0$$. 2. Formula: Use the product rule for derivatives: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$. 3. Identify parts: Let $$u(x) = \exp(x)$$ and $$v(x) = x^2$$. 4. Compute derivatives: $$u'(x) = \exp(x)$$ and $$v'(x) = 2x$$. 5. Apply product rule: $$f'(x) = u'(x)v(x) + u(x)v'(x) = \exp(x) x^2 + \exp(x) 2x = \exp(x)(x^2 + 2x)$$. 6. Final answer: $$\boxed{f'(x) = \exp(x)(x^2 + 2x)}$$.