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Math Help - More Implicit Differentiation

  1. #1
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    More Implicit Differentiation

    x^2+y^2=1
    Find dy/dx and the second derivative.

    The answer says both derivatives equal \frac{-x}{y}.
    The first one does, but the second derivative I got as \frac{y}{x}.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Truthbetold View Post
    x^2+y^2=1
    Find dy/dx and the second derivative.

    The answer says both derivatives equal \frac{-x}{y}.
    The first one does, but the second derivative I got as \frac{y}{x}.
    and how did you get that answer?

    i got neither. my answer for the second derivative was more complicated. i differentiated -x/y with the quotient rule. but you can use the product rule also
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  3. #3
    MHF Contributor kalagota's Avatar
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    aww, i got \frac{-1}{y^3}..
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  4. #4
    GAMMA Mathematics
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    x^2 + y^2 = 1
    2x\frac{dy}{dx} + 2y = 0
    \frac{dy}{dx} = \frac{-y}{x}

    \frac{d^2y}{dx^2} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{dy}{dx} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{-y}{x} = \frac{-1}{x} + \frac{-y^2}{x^3} = \frac{-(x^2+y^2)}{x^3}

    I got something different too. Hmm, did I miss something?
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  5. #5
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by colby2152 View Post
    x^2 + y^2 = 1
    2x\frac{dy}{dx} + 2y = 0
    \frac{dy}{dx} = \frac{-y}{x}

    \frac{d^2y}{dx^2} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{dy}{dx} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{-y}{x} = \frac{-1}{x} + \frac{-y^2}{x^3} = \frac{-(x^2+y^2)}{x^3}

    I got something different too. Hmm, did I miss something?

    it's because, \frac{dy}{dx} = \frac{-x}{y} and not \frac{dy}{dx} = \frac{-y}{x}..
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  6. #6
    Eater of Worlds
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    Quote Originally Posted by colby2152 View Post
    x^2 + y^2 = 1
    2x\frac{dy}{dx} + 2y = 0
    \frac{dy}{dx} = \frac{-y}{x}

    \frac{d^2y}{dx^2} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{dy}{dx} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{-y}{x} = \frac{-1}{x} + \frac{-y^2}{x^3} = \frac{-(x^2+y^2)}{x^3}

    I got something different too. Hmm, did I miss something?
    You're OK, except it should be y^{3} in the denominator.
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  7. #7
    GAMMA Mathematics
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    Quote Originally Posted by galactus View Post
    You're OK, except it should be y^{3} in the denominator.
    I see what I did, I flipped the derivative in the second line. It's easy to miss these things when typing them.
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