(d2xdy2)=ddy(dxdy)=ddy(1dydx)=1(dydx)2xddy(dydx)=−1(dydx)2xd2ydx2dxdy=−(d2ydx2)(dydx)−3

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$\frac{d^{2} x}{d y^{2}}$ Equals:

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$(\frac{d^{2} y}{d x^{2}}) {(\frac{d y}{d x})}^{2}$

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

${(\frac{d^{2} y}{d x^{2}})}^{- 1}$

${(\frac{d^{2} y}{d x^{2}})}^{- 1} {(\frac{d y}{d x})}^{- 3}$

answer is B.

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Detailed Solution

$(\frac{d^{2} x}{d y^{2}}) = \frac{d}{d y} (\frac{d x}{d y}) = \frac{d}{d y} (\frac{\frac{1}{d y}}{d x}) = \frac{1}{{(\frac{d y}{d x})}^{2}} x \frac{d}{d y} (\frac{d y}{d x}) =$ $- \frac{1}{{(\frac{d y}{d x})}^{2}} x \frac{d^{2} y}{d x^{2}} \frac{d x}{d y} = - (\frac{d^{2} y}{d x^{2}}) {(\frac{d y}{d x})}^{- 3}$