\[\frac{d}{dx}(cosech^{-1}x)\]
let \[y=cosech^{-1}x\]
then \[cosechy=x\]
.Differentiating implicitly gives
\[-cosechycothy \frac{dy}{dx}=1 \rightarrow \frac{dy}{dx}=- \frac{1}{cosechy cothy}\]
.To express
\[\frac{dy}{dx}\]
in terms of \[x\]
use \[cosechy=x,\; cothy = \sqrt{cosech^2y+1} =\sqrt{x^2+1}\]
.Then
\[\frac{dy}{dx}=- \frac{1}{x \sqrt{x^2+1}}\]
.