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

$$\begin{array}{cc}y& =3\ln (4x^3+2)\\ \frac{\mathrm{d}y}{\mathrm{d}x}& =3\left(\frac{12x^2}{4x^3+2}\right)\\ & =\frac{36x^2}{4x^3+2}\\ & =\frac{18x^2}{2x^3+1}∎\end{array}$$

$$\begin{array}{cc}& {\displaystyle \int \frac{3x^2-2}{\sqrt{x}}\mathrm{d}x}\\ & ={\displaystyle \int (\frac{3x^2}{\sqrt{x}}-\frac{2}{\sqrt{x}})\mathrm{d}x}\\ & ={\displaystyle \int (3x^{\frac{3}{2}}-2x^{-\frac{1}{2}})\mathrm{d}x}\\ & =3\left(\frac{x^{\frac{5}{2}}}{\frac{5}{2}\left(1\right)}\right)-2\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}\left(1\right)}\right)+c\\ & =\frac{6x^{\frac{5}{2}}}{5}-4x^{\frac{1}{2}}+c∎\end{array}$$