1. **Problem:** Differentiate the following functions with respect to $x$: d) $y = (2x^2 - 5)^9$ e) $y = \frac{(2x + 1)^6}{3}$ g) $y = 6(5 - x)^5$ h) $y = \frac{1}{(2x + 5)^8}$ 2. **Formula and rules:** Use the chain rule for differentiation: If $y = [f(x)]^n$, then $$\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$$ For a constant multiple $c$, $\frac{d}{dx}[c \cdot g(x)] = c \cdot g'(x)$. For $y = \frac{1}{u(x)} = u(x)^{-1}$, use the power rule and chain rule. 3. **Step-by-step differentiation:** d) $y = (2x^2 - 5)^9$ - Inner function: $u = 2x^2 - 5$ - Derivative of inner function: $u' = 4x$ - Apply chain rule: $$\frac{dy}{dx} = 9(2x^2 - 5)^8 \cdot 4x = 36x(2x^2 - 5)^8$$ e) $y = \frac{(2x + 1)^6}{3} = \frac{1}{3}(2x + 1)^6$ - Inner function: $u = 2x + 1$ - Derivative of inner function: $u' = 2$ - Apply chain rule: $$\frac{dy}{dx} = \frac{1}{3} \cdot 6(2x + 1)^5 \cdot 2 = \frac{12}{3}(2x + 1)^5 = 4(2x + 1)^5$$ g) $y = 6(5 - x)^5$ - Inner function: $u = 5 - x$ - Derivative of inner function: $u' = -1$ - Apply chain rule: $$\frac{dy}{dx} = 6 \cdot 5(5 - x)^4 \cdot (-1) = -30(5 - x)^4$$ h) $y = \frac{1}{(2x + 5)^8} = (2x + 5)^{-8}$ - Inner function: $u = 2x + 5$ - Derivative of inner function: $u' = 2$ - Apply chain rule: $$\frac{dy}{dx} = -8(2x + 5)^{-9} \cdot 2 = -16(2x + 5)^{-9} = -\frac{16}{(2x + 5)^9}$$ 4. **Final answers:** d) $\boxed{36x(2x^2 - 5)^8}$ e) $\boxed{4(2x + 1)^5}$ g) $\boxed{-30(5 - x)^4}$ h) $\boxed{-\frac{16}{(2x + 5)^9}}$