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Math Help - Differentials

  1. #1
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    Differentials

    If x^2y + yx^2 = 6, then what does \frac{d^2y}{dx^2} equal at the point (1,3)?


    **I have not attempted this solution because I do not understand how \frac{d^2y}{dx^2} works.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Cursed View Post
    If x^2y + yx^2 = 6, then what does \frac{d^2y}{dx^2} equal at the point (1,3)?


    **I have not attempted this solution because I do not understand how \frac{d^2y}{dx^2} works.
    I am going to assume this was supposed to be
    x^2y + xy^2 = 6

    Start with your first derivative:
    2xy + 2x \frac{dy}{dx} + y^2 + 2xy\frac{dy}{dx} = 0

    so
    \frac{dy}{dx} = -\frac{2xy + y^2}{2x + 2xy}

    Now take the second derivative:
    \frac{d^2y}{dx^2} = -\frac{\left ( 2y + 2x\frac{dy}{dx}  \right ) (2x + 2xy) + (2xy + y^2) \left ( 2 + 2y + 2x \frac{dy}{dx} \right ) }{(2x + 2xy)^2}

    Now plug in your expression for the derivative and simplify.

    -Dan
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  3. #3
    Eater of Worlds
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    Well, we can differentiate implicitly and then do it again.

    x^{2}y+x^{2}y=6

    2(x^{2}y)=6

    x^{2}y=3

    Product rule:

    x^{2}y'+2xy=0

    y'=\frac{-2xy}{x^{2}}=\frac{-2y}{x}

    Do it again using quotient rule:

    y''=\frac{-2xy'+2y}{x^{2}}=\frac{2(y-xy')}{x^{2}}

    But, note what y' is equal to from before. Sub that in:

    y''=\frac{2(y-x(\frac{-2y}{x}))}{x^{2}}=\frac{6y}{x^{2}}

    Plug in your values.
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  4. #4
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    Thank you both for the help. I understand it now.
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