Question

If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to

 

विकल्प

• \[\frac{f' g'' - g'f''}{\left( f' \right)^3}\]

• \[\frac{f' g'' - g'f''}{\left( f' \right)^2}\]

• \[\frac{g''}{f''}\]

• \[\frac{f'' g' - g'' f'}{\left( g' \right)^3}\]

Answer 1

(a) \[\frac{f' g'' - g'f''}{\left( f' \right)^3}\]

Here,
x = f(t) and y = g(t)

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

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

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

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

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