Use implicit differentiation to find d^2y/dx^2 for `sqrt(x*y) = -2 + x^2*y`     

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the second derivative y'' for `sqrt(x*y) = -2 + x^2*y`

Using implicit differentiation, we get

sqrt x * (sqrt y)' + (sqrt x)'*sqrt y = x^2*y' + y*2x

=> y'*sqrt x*0.5/sqrt y + 0.5*sqrt y/sqrt x = x^2*y' + y*2x

=> y'(sqrt x*0.5/sqrt y - x^2) = y*2x - 0.5*sqrt y/sqrt x

=> y' = (y*2x - 0.5*sqrt y/sqrt x)/(sqrt x*0.5/sqrt y - x^2)

y'' = [(y*2x - 0.5*sqrt y/sqrt x)'*(sqrt x*0.5/sqrt y - x^2) - (y*2x - 0.5*sqrt y/sqrt x)*(sqrt x*0.5/sqrt y - x^2)']/(sqrt x*0.5/sqrt y - x^2)^2

=> [(y'2x + 2y - 0.25y'/sqrt(xy) + 0.25*sqrt y/x*sqrt x)*(sqrt x*0.5/sqrt y - x^2) - (y*2x - 0.5*sqrt y/sqrt x)*(-0.25/sqrt xy - 0.25y'*sqrt x/y*sqrt y - 2x)]/ (sqrt x*0.5/sqrt y - x^2)^2

The required second derivative is [(y'2x + 2y - 0.25y'/sqrt(xy) + 0.25*sqrt y/x*sqrt x)*(sqrt x*0.5/sqrt y - x^2) - (y*2x - 0.5*sqrt y/sqrt x)*(-0.25/sqrt xy - 0.25y'*sqrt x/y*sqrt y - 2x)]/ (sqrt x*0.5/sqrt y - x^2)^2

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