1. **Identify the derivatives:** (i) Given $y = (3x^2 + 1)^{\frac{3}{2}}$ Use the chain rule: If $y = [u(x)]^n$, then $\frac{dy}{dx} = n[u(x)]^{n-1} \cdot u'(x)$. Here, $u(x) = 3x^2 + 1$, $n = \frac{3}{2}$. Calculate $u'(x) = 6x$. So, $$\frac{dy}{dx} = \frac{3}{2}(3x^2 + 1)^{\frac{1}{2}} \cdot 6x = 9x(3x^2 + 1)^{\frac{1}{2}}.$$ (ii) Given $y = \frac{x^{2-3}}{x^2} = \frac{x^{-1}}{x^2} = x^{-1 - 2} = x^{-3}$. Derivative of $x^n$ is $nx^{n-1}$. So, $$\frac{dy}{dx} = -3x^{-4} = -\frac{3}{x^4}.$$ (iii) Given $f(q) = \frac{3q^2}{500} - \frac{3q}{125} + 24$. Differentiate term by term: $$f'(q) = \frac{3 \cdot 2q}{500} - \frac{3}{125} + 0 = \frac{6q}{500} - \frac{3}{125} = \frac{3q}{250} - \frac{3}{125}.$$ 2. **Examine the integrals:** (i) $$\int (4x^4 + 5x^3 - 6x + 8) \, dx = \int 4x^4 \, dx + \int 5x^3 \, dx - \int 6x \, dx + \int 8 \, dx.$$ Integrate each term using power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$: $$= 4 \cdot \frac{x^{5}}{5} + 5 \cdot \frac{x^{4}}{4} - 6 \cdot \frac{x^{2}}{2} + 8x + C = \frac{4}{5}x^{5} + \frac{5}{4}x^{4} - 3x^{2} + 8x + C.$$ (ii) Given $$\int \frac{x^{2-3}}{x^2} dx = \int x^{-1 - 2} dx = \int x^{-3} dx.$$ Integrate: $$\int x^{-3} dx = \frac{x^{-3+1}}{-3+1} + C = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^{2}} + C.$$