# Derivative of Inverse Hyperbolic Cosecant

## 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 $y$

$\blacksquare$