1. **Problem 1:** Find the derivative with respect to $x$ of the function $D_x = x(4 - \frac{1}{2}x)$. 2. First, expand the expression inside the derivative: $$D_x = 4x - \frac{1}{2}x^2$$ 3. Now differentiate term by term: $$\frac{d}{dx}(4x) = 4$$ $$\frac{d}{dx} \left(-\frac{1}{2}x^2 \right) = -\frac{1}{2} \cdot 2x = -x$$ 4. Combine the derivatives: $$\frac{dD_x}{dx} = 4 - x$$ --- 5. **Problem 2:** Find the derivative $\frac{dy}{dx}$ if $$y = \frac{x^3 - 27}{x - 3}$$ 6. Notice that the numerator is a difference of cubes: $$x^3 - 27 = (x - 3)(x^2 + 3x + 9)$$ 7. Substitute this factorization: $$y = \frac{(x - 3)(x^2 + 3x + 9)}{x - 3}$$ 8. Cancel the common factor $(x - 3)$ (for $x \neq 3$): $$y = x^2 + 3x + 9$$ 9. Differentiate term by term: $$\frac{dy}{dx} = 2x + 3$$ **Final answers:** 1. $$\frac{dD_x}{dx} = 4 - x$$ 2. $$\frac{dy}{dx} = 2x + 3$$