1. **State the problem:** Find the derivative of the function $y = x(2 - e^x)^3$. 2. **Formula used:** We will use the product rule and the chain rule. - Product rule: $\frac{d}{dx}[u \cdot v] = u'v + uv'$. - Chain rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$. 3. **Identify parts:** Let $u = x$ and $v = (2 - e^x)^3$. 4. **Differentiate $u$:** $u' = \frac{d}{dx}[x] = 1$. 5. **Differentiate $v$ using chain rule:** - Let $w = 2 - e^x$, so $v = w^3$. - Then $v' = 3w^2 \cdot w'$. - Calculate $w' = \frac{d}{dx}[2 - e^x] = 0 - e^x = -e^x$. - So $v' = 3(2 - e^x)^2 \cdot (-e^x) = -3e^x(2 - e^x)^2$. 6. **Apply product rule:** $$ \frac{dy}{dx} = u'v + uv' = 1 \cdot (2 - e^x)^3 + x \cdot \left(-3e^x(2 - e^x)^2\right) $$ 7. **Simplify:** $$ \frac{dy}{dx} = (2 - e^x)^3 - 3xe^x(2 - e^x)^2 $$ This is the derivative of the given function. **Final answer:** $$ \boxed{\frac{dy}{dx} = (2 - e^x)^3 - 3xe^x(2 - e^x)^2} $$