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Homework Help: Derivative of arctanh

  1. Jun 27, 2006 #1
    Yeah, I was working through this problem and it differs from the answer that my friend got.

    Using implicit differentiation, find the derivative of [tex]\mbox{arc}\tanh \frac{x}{2}[/tex] and state the domain for which the derivative applies

    [tex]\begin{align*}
    y = \arctanh \frac{x}{2} \\
    \Leftrightarrow x = 2 \tanh y
    \end{align*}[/tex]

    [tex]\frac{d}{dx}x = \frac{d}{dx}2 \tanh y[/tex]
    [tex]\Rightarrow 1 = 2\ \mbox{sech}^2 y \cdot \frac{dy}{dx}[/tex]
    [tex]\Rightarrow \frac{dy}{dx} = \frac{1}{2\ \mbox{sech}^2 y}[/tex]
    [tex]\Rightarrow \frac{dy}{dx} = \frac{1}{2 - 2 \tanh^2 y}[/tex]
    [tex]\Rightarrow \frac{dy}{dx} = \frac{1}{2 - 2 \frac{x^2}{4}}[/tex]
    [tex]\Rightarrow \frac{dy}{dx} = \frac{1}{2 - \frac{x^2}{2}}[/tex]
     
    Last edited: Jun 27, 2006
  2. jcsd
  3. Jun 27, 2006 #2

    0rthodontist

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    Science Advisor

    The derivative of what now? d(x/2)/dx = 1/2. I assume you mean tanh-1(x/2).

    Yeah, you're right.
     
    Last edited: Jun 27, 2006
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