Question

If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?

Answer 1

Here.

x = f(t) and y = g(t)

\[\Rightarrow \frac{dx}{dt} = f'\left( t \right) \text { and }\frac{dy}{dt} = g'\left( t \right)\]
\[ \therefore \frac{dy}{dx} = \frac{g'\left( t \right)}{f'\left( t \right)}\]

\[\Rightarrow \frac{d^2 y}{d x^2} = \frac{d}{dt} \left\{ \frac{g^{'} \left( t \right)}{f^{'} \left( t \right)} \right\} \times \frac{dt}{dx}\]

\[ = \frac{f^{'}\left( t \right) g^{''} \left( t \right) - g^{'} \left( t \right) f^{''} \left( t \right)}{\left[ f^{'} \left( t \right) \right]^2} \times \frac{1}{f'\left( t \right)}\]

\[ = \frac{f^{'} \left( t \right) g^{''} \left( t \right) - g^{'} \left( t \right) f^{''} \left( t \right)}{\left[ f^{'} \left( t \right) \right]^3}\]