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Math Help - Differentiation

  1. #1
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    Differentiation

    Please can you help, i can't figure it out.

    Show that the derivative of:

    x + y = x/y

    can be written as

    y^3/x^2(2-y)

    The dfferentiation is really simple and i got
    (1-y)/(x + 2y) when i times the left hand side by y before differentiating. However i cannot make it into that form!
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, mathshelpneeded!

    Show that the derivative of: .x + y .= .x/y
    . . can be written as: .y' .= .y/x(2 - y)

    The differentiation is really simple and i got: .y' .= .(1 - y)/(x + 2y) . [1]
    However i cannot make it into that form!
    I'm not sure how to get it into that form naturally,
    . . but I can hammer it into that form . . .


    From the original equation, we have:
    . . xy + y .= .x . . y .= .x(1 - y) .[2] . . 1 - y .= .y/x . [3]

    The original equation is: .x + y .= .x/y
    Add y to both sides: .x + 2y .= .x/y + y .= .(x + y)/y
    . . From [2]: .x + 2y .= .[x + x(1 - y)]/y . . x + 2y .= .x(2 - y)/y . [4]

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y/x . . . . . . . y
    Substitute [3] and [4] into [1]: .y' .= . ------------ .= .-----------
    . . . . . . . . . . . . . . . . . . . . . . . . . . . x(2 - y)/y . . . .x(2 - y)

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