Mathematics - Differentiation Question with Solution | TestHub

$\frac{dx}{d\theta }=2\cos \theta -2\cos 2\theta$

$\frac{dy}{d\theta }=-2\sin \theta +2\sin 2\theta$

$\therefore \frac{dy}{dx}=\frac{\sin 2\theta -\sin \theta }{\cos \theta -\cos 2\theta }$

$=\frac{2\sin \frac{\theta }{2}.\cos \frac{3\theta }{2}}{2\sin \frac{\theta }{2}.\sin \frac{3\theta }{2}}=\cot \frac{3\theta }{2}$

$\frac{d^{2}y}{d{x}^2}=\frac{d}{d\theta }\left(\frac{dy}{dx}\right)\frac{d\theta }{dx}=-\frac{3}{2}{\mathrm{cosec}}^2\frac{3\theta }{2}.\frac{d\theta }{dx}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{-\frac{3}{2}{\mathrm{cosec}}^2\frac{3\theta }{2}}{2\left(\cos \theta -\cos 2\theta \right)}$

$\Rightarrow {\left.\frac{d^{2}y}{d{x}^2}\right|}_{\theta =\pi }=\frac{3}{4\left(-1-1\right)}=\frac{3}{8}$