Given the parametric equations x-ft-y-gt- then d2y-dx2 equals

We have dy/dx=dy/dt/dx/dt ∴d^2y/dx^2=d/dt(dy/dt/dx/dt)dt/dx =d^2y/dt^2(dx/dt) (d^2x/dt^2)dy/dt/(dx/dt)^2×1/dx/dt =(dx/dt).(d^2y/dt^2) (d^2x/dt^2).(dy/dt)/(dx/dt ...

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Standard XII

Mathematics

Parametric Differentiation

Given the par...

Question

Given the parametric equations $x = f (t), y = g (t)$ , then $\frac{d^{2} y}{d x^{2}}$ equals

\frac{\frac{d^{2} y}{d t^{2}} . \frac{d x}{d t} - \frac{d y}{d t} \frac{d^{2} x}{d t^{2}}}{{(\frac{d x}{d t})}^{2}}

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\frac{\frac{d x}{d t} \frac{d^{2} y}{d t^{2}} - \frac{d^{2} x}{d t^{2}} \frac{d y}{d t}}{{(\frac{d x}{d t})}^{3}}

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\frac{\frac{d^{2} y}{d t^{2}}}{\frac{d^{2} x}{d t^{2}}}

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\frac{\frac{d^{2} y}{d t^{2}} . \frac{d x}{d t} - \frac{d y}{d t} \frac{d^{2} x}{d t^{2}}}{(\frac{d x}{d t})}

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Solution

The correct option is B $\frac{\frac{d x}{d t} \frac{d^{2} y}{d t^{2}} - \frac{d^{2} x}{d t^{2}} \frac{d y}{d t}}{{(\frac{d x}{d t})}^{3}}$
We have $\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
$∴ \frac{d^{2} y}{d x^{2}} = \frac{d}{d t} (\frac{\frac{d y}{d t}}{\frac{d x}{d t}}) \frac{d t}{d x}$
$= \frac{\frac{d^{2} y}{d t^{2}} (\frac{d x}{d t}) - (\frac{d^{2} x}{d t^{2}}) \frac{d y}{d t}}{{(\frac{d x}{d t})}^{2}} \times \frac{1}{\frac{d x}{d t}}$
$= \frac{(\frac{d x}{d t}) . (\frac{d^{2} y}{d t^{2}}) - (\frac{d^{2} x}{d t^{2}}) . (\frac{d y}{d t})}{{(\frac{d x}{d t})}^{3}}$ .

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Given the parametric equations x=f(t),y=g(t), then d2ydx2 equals

The correct option is B dxdtd2ydt2−d2xdt2dydt(dxdt)3We have dydx=dydtdxdt ∴d2ydx2=ddt(dydtdxdt)dtdx =d2ydt2(dxdt)−(d2xdt2)dydt(dxdt)2×1dxdt =(dxdt).(d2ydt2)−(d2xdt2).(dydt)(dxdt)3.

Given the parametric equations $x = f (t), y = g (t)$ , then $\frac{d^{2} y}{d x^{2}}$ equals