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Derivative of (tanh^-1(sinh(2x)))

  1. Apr 11, 2014 #1
    1. The problem statement, all variables and given/known data
    Derivative of (tanh^-1(sinh(2x)))


    2. Relevant equations
    see above


    3. The attempt at a solution
    Im trying to use the chain rule here but I cant even get the first step. Can you guide me?
     
  2. jcsd
  3. Apr 11, 2014 #2

    Dick

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    tanh^(-1)(x) is also call arctanh(x). What's the derivative of that? If you don't know you could try and look it up.
     
  4. Apr 11, 2014 #3
    Yes I am aware of that. The derivative is:
    1/(1-x^2)

    But how do I use the chain rule here?
     
  5. Apr 11, 2014 #4

    Dick

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    The chain rule says the derivative of (f(g(x)))'=f'(g(x))*g'(x). f is arctanh and g(x) is cosh(2x). So?
     
  6. Apr 11, 2014 #5

    HallsofIvy

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    Another way to do this is to change [itex]y= tanh^{-1}(sinh(2x)[/itex] to [itex]tanh(y)= sinh(2x)[/itex].

    Differentiate both sides with respect to x (so the left side will be the derivative of tanh(y) with respect to y times dy/dx) and then solve for dy/dx.
     
  7. Apr 12, 2014 #6
    it would be:

    (1/(1-x^2)) * sinh2x * 2cosh2x

    Is that correct? I checked wolfram for the derivative and i got:
    -4cosh(2x) / cosh(4x)-3

    Im not sure how I could get my solution in that format?

    Thank you so much for helping me out
     
    Last edited: Apr 12, 2014
  8. Apr 12, 2014 #7

    Dick

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    If f is arctanh and g(x) is cosh(2x). Then just as f'(x)=1/(1-x^2), f'(g(x)) should be 1/(1-g(x)^2). NOT 1/(1-x^2). Don't worry too much about comparing with Wolfram. There are many different looking ways to write the same answer.
     
  9. Apr 13, 2014 #8
    so is this correct:
    Derivative of (tanh^-1(sinh(2x))) is

    (1/1-(2sinh(2x)^2) * 2cosh(2x) ?
     
  10. Apr 13, 2014 #9

    Dick

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    Ok, so the right hand factor should be g'(x)=(sinh(2x))'=2cosh(2x). That looks right. The rest should be f'(g(x))=arctanh'(g(x))=1/(1-g(x)^2)). That doesn't look right. What's wrong with it?
     
  11. Apr 13, 2014 #10

    oh a careless mistake.
    is it:
    (1/1-(sinh(2x)^2) * 2cosh(2x) ?
     
  12. Apr 13, 2014 #11

    Dick

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    Yes, it is. You should balance your parentheses more carefully, but I know what you mean.
     
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