D 2 x - d y 2 - d 2 y - d x 2 - dx - dy 3

d ^2 xd y^2=d(d xd y)d y=d(1(d yd x))d xd yd x=d xd y× 1(d yd x)^2×d^2 yd x^2= d^2 yd x^2(d xd y)^3 ...

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Standard XII

Mathematics

Higher Order Derivatives

d 2 x / d y 2...

Question

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

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Solution

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

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d2xdy2=−d2ydx2.(dxdy)3

d2xdy2=d(dxdy)dy=d(1(dydx))dxdydx=dxdy×−1(dydx)2×d2ydx2=−d2ydx2(dxdy)3

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

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