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Math Help - Find d/dx of Natural Log of Hyperbolic Function

  1. #1
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    Find d/dx of Natural Log of Hyperbolic Function

    Can someone check my work and tell me what I did wrong?
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    Re: Find d/dx of Natural Log of Hyperbolic Function

    Practice your identities. Don't stop because you have an answer. Play with it. Convert it into something else. See if your answer actually matches the answer in the back of the book. Try to recognize things like double angle formulas.
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    Re: Find d/dx of Natural Log of Hyperbolic Function

    Quote Originally Posted by SlipEternal View Post
    Practice your identities. Don't stop because you have an answer. Play with it. Convert it into something else. See if your answer actually matches the answer in the back of the book. Try to recognize things like double angle formulas.
    a fast check to see if you're dealing with different forms of some trig expression is to plot the difference of the two expressions. If they are the same thing in different guises the difference will be 0, though you might get glitches at domain edges.
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    Re: Find d/dx of Natural Log of Hyperbolic Function

    I will try to simplify more. I'll get back to you.
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    Re: Find d/dx of Natural Log of Hyperbolic Function

    I played with my original answer substituting the correct trigonometric hyperbolic identities and ended up with y' = 2/[sinh^2(x/2)]. However, I don't know how to simplify further to reach the textbook answer of csch(x). Can someone take it from here to find the textbook answer csch(x)?
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    Re: Find d/dx of Natural Log of Hyperbolic Function

    $\displaystyle \begin{align*} y' & = \dfrac{ \text{sech}^2\left( \frac{x}{2} \right) }{ 2 \tanh\left( \frac{x}{2} \right) } \\ & = \dfrac{ \left( \dfrac{ 1 }{ \cosh^2\left( \frac{x}{2} \right) } \right) }{ 2 \dfrac{ \sinh\left( \frac{x}{2} \right) }{ \cosh\left( \frac{x}{2} \right) } } \cdot \dfrac{\cosh^2\left( \frac{x}{2} \right) }{ \cosh^2\left( \frac{x}{2} \right) } \\ & = \dfrac{ 1 }{ 2\sinh\left( \frac{x}{2} \right)\cosh\left( \frac{x}{2} \right) } \\ & = \dfrac{1}{\sinh(x)} \\ & = \text{csch}(x) \end{align*}$
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    Re: Find d/dx of Natural Log of Hyperbolic Function

    Quote Originally Posted by SlipEternal View Post
    $\displaystyle \begin{align*} y' & = \dfrac{ \text{sech}^2\left( \frac{x}{2} \right) }{ 2 \tanh\left( \frac{x}{2} \right) } \\ & = \dfrac{ \left( \dfrac{ 1 }{ \cosh^2\left( \frac{x}{2} \right) } \right) }{ 2 \dfrac{ \sinh\left( \frac{x}{2} \right) }{ \cosh\left( \frac{x}{2} \right) } } \cdot \dfrac{\cosh^2\left( \frac{x}{2} \right) }{ \cosh^2\left( \frac{x}{2} \right) } \\ & = \dfrac{ 1 }{ 2\sinh\left( \frac{x}{2} \right)\cosh\left( \frac{x}{2} \right) } \\ & = \dfrac{1}{\sinh(x)} \\ & = \text{csch}(x) \end{align*}$
    Thank you very much.
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