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Homework Help: Multi variable derivative

  1. Jun 19, 2008 #1
    whats derivative of arctan (xy) with respect to x





    3. The attempt at a solution

    - I feel kind of stupid for this one, but I can't seem to figure out the derivative here. I know with one variable that arctan goes to 1/(1+x^2), but I'm not sure what to do with two variables. please help.
     
  2. jcsd
  3. Jun 19, 2008 #2
    Since you know that [tex]\frac{d}{dx} \arctan x = \frac{1}{1+x^2}[/tex], you have to apply the chain rule.
     
    Last edited: Jun 19, 2008
  4. Jun 19, 2008 #3

    Defennder

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    The more general formulae would be [tex]\frac{d}{dx} \arctan f(x) = \frac{f'(x)}{1+f(x)^2}[/tex]. You have to use implicit differentiation and the chain rule to do this one.
     
  5. Jun 19, 2008 #4
    - I'm still not grasping it. We haven't really done any problems with implicit differentiation like this.
     
  6. Jun 19, 2008 #5
    [tex]y'=\frac{dy}{dx}[/tex]

    [tex]y=\tan^{-1}{(xy)} \rightarrow y=\tan^{-1}f(x)[/tex]

    Let [tex]f(x)=xy \rightarrow f'(x)=xy'+y[/tex]

    [tex]y'=\frac{f'(x)}{1+[f(x)]^2}[/tex]

    Now you take it.
     
    Last edited: Jun 19, 2008
  7. Jun 19, 2008 #6
    try finding derv of
    arctan (5x)
    or
    arctan(10x)
    ..
    or
    arctan(a.x)
    ..
    now substitute your y for a
     
  8. Jun 20, 2008 #7

    tiny-tim

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    Or d(arctan(xy))/dx = d(arctan(xy))/d(xy) times (d(xy)/dx). :smile:
     
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