Question

If $x=a(\cos t+\log \tan \frac{t}{2}),y=a\sin t$, find $\frac{d^{2}y}{d{t}^2}$ and $\frac{d^{2}y}{d{x}^2}.$

Answer 1

$x=a(cost+\log tan\frac{t}{2}),y=asint$

$\frac{dy}{dt}=acost$

$\frac{d^{2}y}{d{t}^2}=-asint$

$\frac{dx}{dt}=a(-sint+\frac{se{c}^2\frac{t}{2}}{2tan\frac{t}{2}})=a\frac{co{s}^{2}t}{sint}$

$\frac{d^{2}y}{d{x}^2}=\frac{d\left(\frac{dy}{dx}\right)}{dx}=\frac{d(\frac{dy}{dt}\times \frac{dt}{dx})}{dt}\times \frac{dt}{dx}$

$\therefore \frac{d^{2}y}{d{x}^2}=\frac{d(acost\times \frac{sint}{aco{s}^{2}t})}{dt}\times \frac{sint}{aco{s}^{2}t}=\frac{d\left(tant\right)}{dt}\times \frac{sint}{aco{s}^{2}t}$

$=se{c}^{2}t\times \frac{sint}{aco{s}^{2}t}$

$\frac{d^{2}y}{d{x}^2}=\frac{sint}{aco{s}^{4}t}$