## Calculus (3rd Edition)

$$y'= \frac{x^{-2}}{|x^{-1}|\sqrt{(x^{-1})^2-1}}=\dfrac{1}{\sqrt{1-x^2}} .$$
Since $y=\csc^{-1}(x^{-1})$, then the derivative $y'$ is given by $$y'= -\frac{1}{|x^{-1}|\sqrt{(x^{-1})^2-1}} (x^{-1})'= \frac{x^{-2}}{|x^{-1}|\sqrt{(x^{-1})^2-1}} .$$ This can be simplified to: $$y'=\dfrac{1}{\sqrt{1-x^2}}$$