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Implicit differentiation: two answers resulted

  1. Mar 6, 2010 #1
    Could anyone explain that I got two different answers for this question: find [tex]dy/dx[/tex] of [tex]\frac{x}{x+y}-\frac{y}{x}=4[/tex].

    1. using quotient rule:
    [tex]\frac{x+y-(1+dy/dx)x}{(x+y)^{2}}-\frac{x\frac{dy}{dx}-y}{x^{2}}=0[/tex]
    [tex]\frac{y}{(x+y)^{2}}-\frac{x}{(x+y)^{2}}\frac{dy}{dx}+\frac{y}{x^{2}}-\frac{1}{x}\frac{dy}{dx}=0[/tex]
    [tex](\frac{x}{(x+y)^{2}}+\frac{1}{x})\frac{dy}{dx}=\frac{y}{(x+y)^{2}}+\frac{y}{x^{2}}[/tex]
    [tex]x(\frac{1}{(x+y)^{2}}+\frac{1}{x^{2}})\frac{dy}{dx}=y(\frac{1}{(x+y)^{2}}+\frac{1}{x^{2}})[/tex]
    [tex]\frac{dy}{dx}=y/x[/tex]

    2. simplify using common denominator before taking derivative:
    [tex]\frac{x^{2}}{x^{2}+xy}-\frac{xy+y^{2}}{x^{2}+xy}=4[/tex]
    [tex]x^{2}-xy-y^{2}=4x^{2}+4xy[/tex]
    [tex]-y^{2}=3x^{2}+5xy[/tex]
    [tex]-2y\frac{dy}{dx}=6x+5y+5x\frac{dy}{dx}[/tex]
    [tex]\frac{dy}{dx}=-\frac{6x+5y}{2y+5x}[/tex]
     
    Last edited: Mar 6, 2010
  2. jcsd
  3. Mar 6, 2010 #2

    nicksauce

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    [tex]
    x^{2}-xy-y^{2}=4x^{2}+4xy
    [/tex]
    [tex]
    y^{2}=3x^{2}+3xy
    [/tex]

    The second line does not follow from the first.
     
  4. Mar 6, 2010 #3
    Sorry, it's suppose to be "5xy"
     
  5. Mar 6, 2010 #4
    and it's suppose to be -y^2. Now I corrected it.
     
  6. Mar 7, 2010 #5
    Help, anyone? i still don't get why.
     
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