1. **Problem:** Given $x = (2 - y)^5$, find the expression for $5x \frac{dy}{dx} + 2$. 2. **Step 1: Differentiate both sides with respect to $x$.** Since $x$ is expressed in terms of $y$, we use implicit differentiation. Differentiate $x$ with respect to $x$ (which is 1), and differentiate the right side using the chain rule: $$\frac{d}{dx} x = \frac{d}{dx} (2 - y)^5$$ $$1 = 5(2 - y)^4 \cdot \frac{d}{dx} (2 - y)$$ 3. **Step 2: Differentiate inside the chain rule.** $$\frac{d}{dx} (2 - y) = 0 - \frac{dy}{dx} = -\frac{dy}{dx}$$ So, $$1 = 5(2 - y)^4 (-\frac{dy}{dx}) = -5(2 - y)^4 \frac{dy}{dx}$$ 4. **Step 3: Solve for $\frac{dy}{dx}$.** $$\frac{dy}{dx} = -\frac{1}{5(2 - y)^4}$$ 5. **Step 4: Substitute $x = (2 - y)^5$ into the expression $5x \frac{dy}{dx} + 2$.** $$5x \frac{dy}{dx} + 2 = 5(2 - y)^5 \left(-\frac{1}{5(2 - y)^4}\right) + 2$$ 6. **Step 5: Simplify the expression.** $$= (2 - y)^5 \cdot \cancel{5} \cdot \left(-\frac{1}{\cancel{5}(2 - y)^4}\right) + 2 = (2 - y)^{5 - 4} (-1) + 2 = -(2 - y) + 2$$ $$= 2 - (2 - y) = 2 - 2 + y = y$$ **Final answer:** $5x \frac{dy}{dx} + 2 = y$ --- **Summary:** The expression simplifies to $y$.