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Math Help - Implicit Differentiation

  1. #1
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    Implicit Differentiation

    Use implicit differentiation to find dy/dx for xy2 - yx2 = 3xy.

    My working:

    x.(2y.dy/dx)+y^2(1)-y(2x)+dy/dx(x^2)=3x dy/dx + 3y

    ==> 2xy dy/dx + y^2 - 2xy + dy/dx (x^2) = 3x dy/dx + 3y


    Now Im very confused. Do we need to divide both sides, in order to separate dy/dx? Any helpful tips/suggestions would be appreciated

    Thanks!

    Last edited by mr fantastic; June 5th 2010 at 05:47 PM. Reason: Edited post title.
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  2. #2
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    So xy^2 - x^2y = 3xy

    Your working looks right although it's a little messy and I think youre dropping a sign somewhere so allow me:

    y^2 + 2xy\frac{dy}{dx} - x^2\frac{dy}{dx} - 2xy = 3x\frac{dy}{dx} + 3y

    If we throw \frac{dy}{dx}'s to one side and the rest to the other side:

    y^2  - 2xy - 3y =  3x\frac{dy}{dx} - 2xy\frac{dy}{dx} + x^2\frac{dy}{dx}

    Take out \frac{dy}{dx} as factor on RHS

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

    Divide both sides by (3x - 2xy + x^2) and you end up with:

    \frac{y^2  - 2xy - 3y}{3x - 2xy + x^2} =  \frac{dy}{dx}
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  3. #3
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    Quote Originally Posted by Silver View Post
    So xy^2 - x^2y = 3xy
    Im sorry, I didn't type the question in proper notation. it's actually:
    xy^2-yx^2=3xy

    could you please help me with that? I'm still a little confused.

    Thanks!
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  4. #4
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    Quote Originally Posted by spoc21 View Post
    Im sorry, I didn't type the question in proper notation. it's actually:
    xy^2-yx^2=3xy

    could you please help me with that? I'm still a little confused.

    Thanks!
    It shouldn't make a difference
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  5. #5
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    Quote Originally Posted by Silver View Post

    y^2 + 2xy\frac{dy}{dx}<b> - x^2</b>\frac{dy}{dx} - 2xy = 3x\frac{dy}{dx} + 3y
    I actually got + x^2\frac{dy}{dx}, using the product rule/chain rule of derivative.

    is mine incorrect?
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  6. #6
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    That was the sign I mentioned you dropped earlier.

    Look at it this way:

     xy^2 - x^2y = 3xy


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

    y^2 + 2xy\frac{dy}{dx} - x^2\frac{dy}{dx} - 2xy = 3x\frac{dy}{dx} + 3y


    Does that make sense?
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  7. #7
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    yes, makes perfect sense, Thank you!
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