2024 ASRJC Promo Q2

It is given that \(y=\tan 3x\).

1. Show that \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=Ay\frac{\mathrm{d}y}{\mathrm{d}x}\), where \(A\) is a constant to be determined.[3]
2. Hence show that \(\frac{{{\mathrm{d}}^{4}}y}{\mathrm{d}{{x}^{4}}}=B\frac{\mathrm{d}y}{\mathrm{d}x}\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}+Cy\frac{{{\mathrm{d}}^{3}}y}{\mathrm{d}{{x}^{3}}}\), where \(B\) and \(C\) are constants to be determined.[3]