D^2x/dy^2=d/dy(dx/dy )=d/dx(dx/dy )dx/dy =d/dx(1/dy/dx )dx/dy= 1/(dy/dx )^2.d^2y/dx^2.1/dy/dx= 1/(dy/dx )^3d^2y/dx^2 ...

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Higher Order Derivatives

d2x/dy2 equal...

Question

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

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

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

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

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

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Solution

The correct option is C
$- (\frac{d^{2} y}{d x^{2}}) {(\frac{d y}{d x})}^{- 3}$

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

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