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Math Help - Derivative of Inverse of Hyperbolic

  1. #1
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    Derivative of Inverse of Hyperbolic

    Show that \frac{d}{dx}(sinh^{-1}\sqrt{(x^2-1)}) = \frac{1}{\sqrt{(x^2-1)}}
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    Re: Derivative of Inverse of Hyperbolic

    What have you tried so far?
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  3. #3
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    Re: Derivative of Inverse of Hyperbolic

    Quote Originally Posted by Ackbeet View Post
    What have you tried so far?
    Sorry Solved it now. Sorry

    sinh y = \sqrt{(x^2-1)}
    cosh y \frac{dy}{dx} = \frac{x}{\sqrt{x^2-1}}
    \frac{dy}{dx} = \frac{x}{\sqrt{x^2-1}\sqrt{1+(x^2-1)}} I originally didn't see 1 + (x^2-1) = x^2
    \frac{dy}{dx} = \frac{x}{\sqrt{x^2-1}\sqrt{x^2}}
    \frac{dy}{dx} = \frac{x}{x\sqrt{x^2-1}}
    \frac{dy}{dx} = \frac{1}{\sqrt{x^2-1}}
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    Re: Derivative of Inverse of Hyperbolic

    Hmm, interesting. I actually don't agree with the result unless x is non-negative. If that was assumed in the problem statement, great. Otherwise, you can't simplify

    \sqrt{x^{2}}=x. You can say that \sqrt{x^{2}}=|x|. So you could simplify as follows:

    \frac{x}{|x|\sqrt{x^{2}-1}}=\frac{\text{sgn}(x)}{\sqrt{x^{2}-1}},

    where sgn is the signum function.

    Otherwise, your working is correct, I think.
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    Re: Derivative of Inverse of Hyperbolic

    thank you...if so how do i approach the question?
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    Re: Derivative of Inverse of Hyperbolic

    Just present the proof you had in Post # 3, with the modifications mentioned in Post # 4 and Post # 4's answer. If the problem originally stated that x is positive, you can simplify the signum to the expression in the OP. Otherwise, I think you have to leave it in there. To be on the safe side, include a sentence explaining what you're doing.
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  7. #7
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    Re: Derivative of Inverse of Hyperbolic

    Quote Originally Posted by BabyMilo View Post
    Show that \frac{d}{dx}(sinh^{-1}\sqrt{(x^2-1)}) = \frac{1}{\sqrt{(x^2-1)}}
    You could always convert the inverse hyperbolic sine function to its logarithmic equivalent...

    \displaystyle \begin{align*} y &= \sinh^{-1}{(x)} \\ \sinh{(y)} &= x \\ \frac{e^y - e^{-y}}{2} &= x \\ e^y - \frac{1}{e^y} &= 2x \\ e^{2y} - 1 &= 2x\,e^y \\ e^{2y} - 2x\,e^y - 1 &= 0 \\ e^{2y} - 2x\,e^y + \left(-x\right)^2 - \left(-x\right)^2 - 1 &= 0 \\ \left(e^y - x\right)^2 - x^2 - 1 &= 0 \\ \left(e^y - x\right)^2 &= x^2 + 1 \\ e^y - x &= \pm \sqrt{x^2 + 1} \\ e^y &= x \pm \sqrt{x^2 + 1} \\ y &= \ln{\left(x \pm \sqrt{x^2 + 1}\right)}  \end{align*}

    Of course, since \displaystyle \sqrt{x^2 + 1} > x for all \displaystyle x, that means we can only accept \displaystyle y = \ln{\left(x + \sqrt{x^2 + 1}\right)}.


    Therefore

    \displaystyle \begin{align*}y &= \sinh^{-1}{\left(\sqrt{x^2 - 1}\right)} \\ y &= \ln{\left(\sqrt{x^2 - 1} + \sqrt{\left(\sqrt{x^2 - 1}\right)^2 + 1}\right)} \\ y &= \ln{\left(\sqrt{x^2 - 1} + \sqrt{x^2}\right)}\end{align*}

    You can now find the derivative using the chain rule, or exponentiating both sides and using implicit differentiation.
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