Differentiation: JPJC 2024 Promo Q1

(a) $$\begin{aligned} \frac{\mathrm d}{\mathrm dx}\cos^{-1}(3x^2)&=-\frac{6x}{\sqrt{1-(3x^2)^2}}\\ &=-\frac{6x}{\sqrt{1-9x^4}}. \end{aligned}$$

(b) $$\begin{aligned} \frac{\mathrm d}{\mathrm dx}\ln\left(\frac{\sqrt{2x+1}}{x^3}\right) &=\frac{\mathrm d}{\mathrm dx}\left[\frac12\ln(2x+1)-3\ln x\right]\\ &=\frac{2}{2(2x+1)}-\frac3x\\ &=\frac1{2x+1}-\frac3x\\ &=\frac{-5x-3}{x(2x+1)}. \end{aligned}$$

Alternative (not recommended) $$\begin{aligned} \frac{\mathrm d}{\mathrm dx}\ln\left(\frac{\sqrt{2x+1}}{x^3}\right) &=\frac{x^3}{\sqrt{2x+1}}\cdot \frac{x^3\left(\dfrac{2}{2\sqrt{2x+1}}\right)-3x^2\sqrt{2x+1}}{x^6}\\ &=\frac{x^5}{\sqrt{2x+1}}\cdot \frac{x-3(2x+1)}{x^6\sqrt{2x+1}}\\ &=\frac{-5x-3}{x(2x+1)}. \end{aligned}$$