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A particle is moving along the curve whose equation is (xy^3)/(1+y^2)= 8/5. Assume the x-coordinate is increasing at the rate of 6 units/second when the particle is at the point (1,2). At what rate is the y-coordinate of the point changing at that instant? Is it rising or falling?

  • Calculus - ,

    Use implicit differentiation:

    xy^3 / (1+y^2) = 8/5

    (y^3 + 3xy^2 y')(1+y^2) - xy^3 (2yy') = 0

    It's all over (1+y^2)^2, but that can be ignored, since it's never 0.

    y'(3xy^2 + 3xy^4 - 2xy^4) = -y^3(1 + y^2)

    y' = -y^3 (1+y^2)/(3xy^2 + xy^4)

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

    Take it from there.

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