Solution — 2025 Question 3 Differentiation & Applications| Integration & Applications

(a)

\begin{matrix} {(3 \sqrt{x} + \frac{1}{\sqrt{3 x}})}^{2} \\ = 9 x + 2 (3 \sqrt{x}) (\frac{1}{\sqrt{3 x}}) + \frac{1}{3 x} \\ = 9 x + \frac{6 \sqrt{x}}{\sqrt{3 x}} + \frac{1}{3 x} \\ = 9 x + 2 + \frac{1}{3 x} \\ = 9 x + 2 + \frac{1}{3} x^{- 1} \end{matrix}

expand and convert to power form for differentiation

\begin{matrix} \frac{d}{d x} (9 x + 2 + \frac{1}{3} x^{- 1}) \\ = 9 - \frac{1}{3} x^{- 2} \\ = 9 - \frac{1}{3 x^{2}} ∎ \end{matrix}

differentiate the expression

(b)

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

integrate the expression