Solution — 2018 Question 2 Differentiation & Applications| Integration & Applications

(a)

\begin{matrix} y & = 3 \ln (4 x^{3} + 2) \\ \frac{d y}{d x} & = 3 (\frac{12 x^{2}}{4 x^{3} + 2}) \\ = \frac{36 x^{2}}{4 x^{3} + 2} \\ = \frac{18 x^{2}}{2 x^{3} + 1} ∎ \end{matrix}

(b)

\begin{matrix} \int \frac{3 x^{2} - 2}{\sqrt{x}} d x \\ = \int (\frac{3 x^{2}}{\sqrt{x}} - \frac{2}{\sqrt{x}}) d x \\ = \int (3 x^{\frac{3}{2}} - 2 x^{- \frac{1}{2}}) d x \\ = 3 (\frac{x^{\frac{5}{2}}}{\frac{5}{2} (1)}) - 2 (\frac{x^{\frac{1}{2}}}{\frac{1}{2} (1)}) + c \\ = \frac{6 x^{\frac{5}{2}}}{5} - 4 x^{\frac{1}{2}} + c ∎ \end{matrix}