Solution — 2017 Question 2 Differentiation & Applications| Integration & Applications

(a)

$\begin{matrix} y & = \frac{1}{\sqrt{5 x - 2}} \\ = {(5 x - 2)}^{- \frac{1}{2}} \\ \frac{d y}{d x} & = - \frac{1}{2} {(5 x - 2)}^{- \frac{3}{2}} (5) \\ = - \frac{5}{2} {(5 x - 2)}^{- \frac{3}{2}} \\ = - \frac{5}{2 {(5 x - 2)}^{\frac{3}{2}}} ∎ \end{matrix}$

(b)

$\begin{matrix} y & = \frac{{(2 x^{2} - 1)}^{2}}{x^{2}} \\ = \frac{4 x^{4} - 4 x^{2} + 1}{x^{2}} \\ = 4 x^{2} - 4 + \frac{1}{x^{2}} \\ = 4 x^{2} - 4 + x^{- 2} \end{matrix}$

$\begin{matrix} \int \frac{{(2 x^{2} - 1)}^{2}}{x^{2}} d x \\ = \int (4 x^{2} - 4 + x^{- 2}) d x \\ = 4 (\frac{x^{3}}{3 (1)}) - 4 x + \frac{x^{- 1}}{- (1)} + c \\ = \frac{4 x^{3}}{3} - 4 x - \frac{1}{x} + c ∎ \end{matrix}$