Question

If $ x= \phi \left(t\right), y=\Psi\left(t\right) $ , 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 ''} $

Answer 1

Answer: (B)

We have, $x=\phi(t), y=\psi(t)$
$\therefore \,\,\, \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi'}{\phi'}$
$\Rightarrow \,\,\, \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{\psi'}{\phi'}\right)=\frac{d}{d t}\left(\frac{\Psi'}{\phi'}\right) \frac{d t}{d x}$
$=\frac{\phi' \Psi''-\psi' \phi''}{\left(\phi'\right)^{2}} \cdot \frac{1}{\phi'}=\frac{\phi' \Psi''-\psi' \phi''}{\left(\phi'\right)^{3}}$