# `xy - 1 = 2x + y^2` Find the second derivative implicitly in terms of `x` and `y`.

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gsarora17 | Certified Educator

`xy-1=2x+y^2`

Differentiating both sides with respect to x,

`xy'+y=2+2y*y'`

`(x-2y)y'=2-y`

`y'=(2-y)/(x-2y)`

``Differentiating again with respect to x,

`(d^2y)/dx^2=((x-2y)d/dx(2-y)-(2-y)d/dx(x-2y))/(x-2y)^2`

`(d^2y)/dx^2=((x-2y)(-y')-(2-y)(1-2y'))/(x-2y)^2`

`(d^2y)/dx^2=(-xy'+2yy'-2+4y'+y-2yy')/(x-2y)^2`

`(d^2y)/dx^2=((4-x)y'+y-2)/(x-2y)^2`

Now plug in the value of y' in second derivative

`(d^2y)/dx^2=(((4-x)(2-y))/(x-2y)+y-2)/(x-2y)^2`

`(d^2y)/dx^2=((4-x)(2-y)+(x-2y)(y-2))/(x-2y)^3`

`(d^2y)/dx^2=(8-4y-2x+xy+xy-2x-2y^2+4y)/(x-2y)^3`

`(d^2y)/dx^2=(8-4x+2xy-2y^2)/(x-2y)^3`