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