Q. If $x = \phi(t),y = \psi(t),\ then\frac{d^{2}y}{dx^{2}}\$is equal to
A) $\frac{\phi^{'}\psi^{''} - \psi^{'}\phi^{''}}{\left( \phi^{'} \right)^{2}}$
B) $\frac{\phi^{'}\psi^{''} - \psi^{'}\phi^{''}}{\left( \phi^{'} \right)^{3}}$
C) $\frac{\phi^{''}}{\psi^{''}}$
D) $\frac{\psi^{''}}{\phi^{''}}$
Explanation
(b)
Given, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\psi^{'}}{\phi^{'}}$
$\Rightarrow \ \ \ \frac{d^{2}y}{dx^{2}} = \frac{d}{dt}\left( \frac{\psi^{'}}{\phi^{'}} \right)\frac{dt}{dx}$
$= \ \frac{\phi^{'}\psi^{''} - \psi^{'}\phi^{''}}{\left( \phi^{'} \right)^{2}}.\frac{1}{\phi^{'}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.