# Find `(d^2y)/(dx^2)` if `x^2-2y^2=3`

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The variables x and y are related by `x^2-2y^2=3` .

Taking the derivative of both the sides with respect to x,

`2x - 4y*(dy/dx) = 0`

=> `dy/dx = x/(2y)`

The second derivative `(d^2y)/(dx^2) = (2y - 2*x*(dy/dx))/(4y^2)`

= `(2y - 2x*(x/(2y)))/(4y^2)`

= `(2y - x^2/y)/(4y^2)`

= `1/(2y) - x^2/(4*y^3) `

**The required derivative `(d^2y)/(dx^2) = 1/(2y) - x^2/(4*y^3)` **

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Differentiate with respect to x (implicitly)

`2x-4y(dy)/(dx)=0`

`(dy)/(dx)=(-2x)/(-4y)`

`(dy)/(dx)=(1/2)xy^(-1)` (ii)

differentate again w.r.t x

`(d^2y)/(dx^2)=(1/2){y^(-1)-xy^(-2)(dy)/(dx)}`

`=(1/2){y^(-1)-xy^(-2)((xy^(-1))/2)}`

`=(1/2){y^(-1)-(1/2)x^2y^(-3)}`

`=(1/(4y)){2-x^2y^(-2)}`

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`x^2-2y^2=3`

expliciting respect y:

`y=sqrt((x^2-3)/2)=sqrt(2)/2sqrt(x^2-3)`

`y'=sqrt(2)/2x/sqrt(x^2-3)`

`y''=sqrt(2)/2(sqrt(x^2-3)-(x^2)/sqrt(x^2-3))/(x^2-3)=sqrt(2)/2(x^2-3-x^2)/(x^2-3)^(3/2)=-(3sqrt(2))/(2(x^2-3)^(3/2))`

Black line: y(x)

Red line: y'(x)

Blue line: y''(x)