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Math Help - (xy^2-lny-(x+y)/x=0)'

  1. #1
    Boo
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    (xy^2-lny-(x+y)/x=0)'

    Please, what is a derivative of:
    (xy^2-lny-\frac{x+y}{x}=0)' ?
    Many thanks!
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  2. #2
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    Re: (xy^2-lny-(x+y)/x=0)'

    What are you differentiating with respect to? Do you know what a partial derivative is?
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  3. #3
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    Re: (xy^2-lny-(x+y)/x=0)'

    Hello, Boo!

    xy^2- \ln y-\frac{x+y}{x}\:=\:0
    \text{Find }y'

    Multiply by x\!:\;x^2y^2 - x\ln y - x - y \:=\:0

    Differentiate implicitly:

    . . x^2\!\cdot\!2yy' + 2x\!\cdot\!y^2 - x\!\cdot\!\frac{y'}{y} + 1\!\cdot\!\ln y - 1 - y' \:=\:0

    . . 2x^2yy' + 2xy^2 - \frac{x}{y}y' - \ln y - 1 - y' \:=\:0

    . . 2x^2yy' - \frac{x}{y}y' - y' \:=\:1 - 2xy^2 + \ln y

    . . \left(2x^2y - \frac{x}{y} - 1\right)y' \:=\:1-2xy^2+ \ln y

    Multiply by y\!:\;\left(2x^2y^2 - x - y\right)y' \:=\:y(1-2xy^2+ \ln y)

    Therefore: . y' \;=\;\frac{y(1-2xy^2+ \ln y)}{2x^2y^2 - x - y}
    Thanks from Boo
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  4. #4
    Boo
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    Re: (xy^2-lny-(x+y)/x=0)'

    Please, can u explain me first line, differentiate implicitly?
    Many thanks!
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  5. #5
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    Re: (xy^2-lny-(x+y)/x=0)'

    Hello, Boo!

    Please, can u explain me first line, "differentiate implicitly"? . No!

    If you do not understand Implicit Differentiation,
    . . you have no business attempting this problem.

    Thanks from Shakarri
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