Show that \(\frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{\ln 2x}{3x^2} \right) = \frac{1}{3x^3} -

Question

Show that $\frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{\ln 2x}{3x^2} \right) = \frac{1}{3x^3} - \frac{2\ln 2x}{3x^3}$.[3] Integrate $\frac{\ln 2x}{x^3}$ with respect to $x$.[3] Given that the curve $y = \mathrm{f}(x)$ passes through the point $ \left(\frac{1}{2}, 1 \right)$ and is such that $\mathrm{f}'(x) = \frac{\ln 2x}{x^3}$, find $\mathrm{f}(x)$.[3]

Accepted Answer

(b) $ - \frac{1}{4x^2} - \frac{\ln{2x}}{2x^2} + c$ (c) $y = 2 - \frac{1}{4x^2} - \frac{\ln{2x}}{2x^2} $

2024 CGSS P1 Q12

2024 CGSS P1 Q12