1. **State the problem:** Differentiate the function $f(x) = x e^{-x}$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Identify parts:** Here, $u(x) = x$ and $v(x) = e^{-x}$. 4. **Differentiate each part:** - $u'(x) = 1$ - $v'(x) = \frac{d}{dx} e^{-x} = -e^{-x}$ (using chain rule) 5. **Apply product rule:** $$f'(x) = 1 \cdot e^{-x} + x \cdot (-e^{-x}) = e^{-x} - x e^{-x}$$ 6. **Factor the result:** $$f'(x) = e^{-x}(1 - x)$$ **Final answer:** $$\boxed{f'(x) = e^{-x}(1 - x)}$$