Find dy/dx by implicit differentiation. xy=sqrt{x^2+y^2}

implicit
xysqrtx2y2

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Find dy/dx by implicit differentiation. xy=sqrt{x^2+y^2}

Answer:

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Square both sides: x^2y^2 = x^2 + y^2

product rule on left: x^2 (2y y’) + 2xy^2 = 2x + 2y y’

rearrange and factor out y’: (2x^2y - 2y) y’ = 2x - 2xy^2

divide: y’ = (2x - 2xy^2) / (2x^2y - 2y)

simplify: [x(1 - y^2)] / [y( x^2 - 1)]