1. **State the problem:** Find the derivative of the function $f(x) = xe^x$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $u(x) = x$ and $v(x) = e^x$. 3. **Differentiate each part:** - Derivative of $u(x) = x$ is $u'(x) = 1$. - Derivative of $v(x) = e^x$ is $v'(x) = e^x$. 4. **Apply the product rule:** $$f'(x) = 1 \cdot e^x + x \cdot e^x = e^x + xe^x$$ 5. **Factor the expression:** $$f'(x) = e^x(1 + x)$$ **Final answer:** $$\boxed{f'(x) = e^x(1 + x)}$$