Derivative of Inverse Hyperbolic Cosecant

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Theorem

Let $x \in \R_{\ne 0}$.

Let $\operatorname{csch}^{-1} x$ be the inverse hyperbolic cosecant of $x$.


Then:

$\dfrac {\mathrm d}{\mathrm d x} \left({\operatorname{csch}^{-1} x}\right) = \dfrac {-1} {\left\vert{x}\right\vert \sqrt{1 + x^2} }$


Proof

\(\displaystyle y\) \(=\) \(\displaystyle \operatorname{csch}^{-1} x\)
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle \operatorname{csch} y\) where $y \ne 0$
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d x} {\mathrm d y}\) \(=\) \(\displaystyle -\operatorname{csch} y \ \coth y\) Derivative of Hyperbolic Secant Function
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle \dfrac {-1} {\operatorname{csch} y \ \coth y}\) Derivative of Inverse Function
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle \frac {-1} {\operatorname{csch} y \ \sqrt {1 + \operatorname{csch}^2 y} }\) Difference of Squares of Hyperbolic Cotangent and Cosecant
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d}{\mathrm d x} \left({\operatorname{csch}^{-1} x}\right)\) \(=\) \(\displaystyle \frac {-1} {\left\vert{x}\right\vert \sqrt{1 + x^2} }\) Definition of $x$ and $y$


$\blacksquare$


Sources