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Math Help - Inverse tan and inverse of tanh

  1. #1
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    Inverse tan and inverse of tanh

    Hello,

    I know what the inverse of tan is when differentiated, but would the inverse of tanh differentiated follow the same format?

    -1/(1-x^2)

    Thank you!
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  2. #2
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    Let's see...

    \displaystyle y = \tanh{x}

    \displaystyle = \frac{\sinh{x}}{\cosh{x}}


    \displaystyle \frac{dy}{dx} = \frac{\cosh{x}\,\frac{d}{dx}(\sinh{x}) - \sinh{x}\,\frac{d}{dx}(\cosh{x})}{\cosh^2{x}}

    \displaystyle = \frac{\cosh{x}\cosh{x} - \sinh{x}\sinh{x}}{\cosh^2{x}}

    \displaystyle = \frac{\cosh^2{x} - \sinh^2{x}}{\cosh^2{x}}

    \displaystyle = \frac{1}{\cosh^2{x}}

    \displaystyle = \textrm{sech}^2{x}.


    This is almost identical to \displaystyle \frac{d}{dx}(\tan{x}) = \sec^2{x}
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  3. #3
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    And therefore, with x= tanh(y),
    d(tanh^{-1}(x))}{dx}= \frac{1}{sech^2(y)}= \frac{1}{1- tanh^2(y)}= \frac{1}{1- x^2}
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