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Math Help - differentiation of sinh function

  1. #1
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    differentiation of sinh function

    I'm trying to differentiate sinh^(-1) (KL). (so arsinh(KL)) with respect to K and also with respect to L. Can some one point me in the right direction? Thanks
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  2. #2
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    Hello, willowtree!

    Differentiate: . y \:=\:\sinh^{-1}(kx) with respect to x.
    Take \sinh of both sides: . \sinh (y) \;=\;kx

    Differentiate implicitly: . \cosh(y)\,\frac{dy}{dx} \:=\:k \quad\Rightarrow\quad \frac{dy}{dx} \:=\:\frac{k}{\cosh(y)}


    We have: . \cosh^2(y) - \sinh^2(y) \:=\:1 \quad\Rightarrow\quad \cos^2(y) \:=\:1 + \sin^2(y) \;=\;1 + (kx)^2

    . . Hence: . \cos(y) \:=\:\sqrt{1+k^2x^2}

    Therefore: . \frac{dy}{dx} \;=\;\frac{k}{\sqrt{1+k^2x^2}}

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