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Math Help - need help dervative of hyperbolic

  1. #1
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    need help dervative of hyperbolic

    Find d/dx [arcsinh (x)]
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  2. #2
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    the derivative of archsinh(x) is  \dfrac{1} {\sqrt{x^{2}+1}}
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  3. #3
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    Hello, gracy!

    Viet is correct . . . There is a formula for it.


    Find \frac{d}{dx}\left[\text{arcsinh}(x)\right]
    If we are expected to derive that formula, we need to know a few facts:

    . . \frac{d}{dx}\left(\sinh(y)\right) \:=\:\cosh(y) [1]

    . . \cosh^2(y) - \sinh^2(y)\:=\:1\quad\Rightarrow\quad\cosh(y) \:=\:\sqrt{1 + \sinh^2(y)} [2]


    We have: . y \;=\;\text{arcsinh}(x)

    Then: . \sinh(y) \;=\;x [3]


    Using [1], differentiate implicitly: . \cosh(y)\cdot\frac{dy}{dx} \;=\;1

    . . Then we have: . \frac{dy}{dx}\:=\:\frac{1}{\cosh(y)}

    From [2], we have: . \frac{dy}{dx}\:=\:\frac{1}{\sqrt{1 + \sinh^2(y)}}

    From [3], we have: . \frac{dy}{dx}\:=\:\frac{1}{\sqrt{1 + x^2}} . . . ta-DAA!

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