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Differentiation Of Inverse Function

  1. Aug 3, 2008 #1

    Air

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    1. The problem statement, all variables and given/known data
    [itex]\frac{\mathrm{d}\left(\frac{1}{a}\tan ^{-1} \left(\frac{x}{a}\right)\right)}{\mathrm{d}x}[/itex]


    2. The attempt at a solution
    Let [itex]y = \frac{1}{a}\tan ^{-1} \left(\frac{x}{a}\right)[/itex]
    [itex]\therefore x = a \tan \left(ay\right)[/itex]
    Differentiate with respect to [itex]x[/itex] [itex]\rightarrow 1 = a \sec ^2 \left(ay\right) \frac{\mathm{d}y}{\mathrm{d}x}[/itex]
    [itex]\therefore \frac{\mathm{d}y}{\mathrm{d}x} = \frac{1}{a \sec ^2 \left(ay\right)} = \frac{1}{a \tan ^2 \left(ay\right) + a}[/itex]
    [itex]x = a \tan \left(ay\right) \therefore \frac{1}{a \tan ^2 \left(ay\right) + a} = \frac{1}{\left(a \tan \left(ay\right)\right)^2 + a} = \frac{1}{x^2 + a}[/itex]


    3. The problem I encountered
    However, thee answer is incorrect. The correct answer is:
    [itex]= \frac{1}{a^2+x^2}[/itex]
    Where have I gone wrong?
     
  2. jcsd
  3. Aug 3, 2008 #2
    x=a tan(ay)
    1= a sec^2(ay) d(ay)/dx
     
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