Solution — 2020 Question 2 Differentiation & Applications| Integration & Applications

(a)

\begin{matrix} y & = e^{3 x + 2} \\ \frac{d y}{d x} & = (3) e^{3 x + 2} \\ = 3 e^{3 x + 2} ∎ \end{matrix}

(b)

\begin{matrix} y & = 3 \ln (4 + 2 x^{3}) \\ \frac{d y}{d x} & = 3 (\frac{6 x^{2}}{4 + 2 x^{3}}) \\ = \frac{18 x^{2}}{4 + 2 x^{3}} \\ = \frac{9 x^{2}}{2 + x^{3}} ∎ \end{matrix}

(c)

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