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How to find derivative of tanh^-1(x)

  1. Apr 11, 2014 #1
    how do I find the derivative of tanh^-1(sinh(2x))

    do I just find derivative of tanh^-1(x) this then substitute sinh(2x) into x??
     
  2. jcsd
  3. Apr 11, 2014 #2

    Erland

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    Science Advisor

    Yes, but since the Chain Rule applies here, you must not forget the inner derivative.

    It is also possible that this can be rewritten and simplified, but I'm too tired to figure this out now...
     
  4. Apr 13, 2014 #3
    Chain Rule, Chain Rule, Chain Rule!
    [itex] \frac{d}{dx} (\tanh^{-1} \sinh 2x) = \frac{d}{dx} (\tanh^{-1} \sinh 2x) \cdot \frac{d}{dx} \sinh 2x = \frac{1}{1 - \sinh^2 2x} \cdot 2\cosh2x [/itex]
    Thus,
    [itex] \frac{d}{dx} (\tanh^{-1} \sinh 2x) = \frac{2 \cosh 2x}{1 - \sinh^2 2x} [/itex]
     
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