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Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.
x^2y + y^2x = 3

  • Math -

    d[x^2y + y^2x] = 0 -->

    2xydx + x^2dy + 2xydy + y^2dx = 0 ---->

    (2xy + y^2)dx + (2xy + x^2) dy = 0 ---->

    dy/dx = -(2xy + y^2)/(2xy + x^2)

  • Math -

    use the product rule for each term on the left side
    x^2(dy/dx) + y(2x)(dx/dx) + y^2(dx/dx) + x(2y)(dy/dx) = 0
    x^2(dy/dx) + 2xy + y^2 + 2xy(dy/dx) = 0
    dy/dx is a common factor, so ...

    dy/dx(x^2 + 2xy) = -2xy - y^2
    dy/dx = (-2xy - y^2)/(x^2 + 2xy)

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