Answer
\[\frac{dy}{dx}=\frac{-1}{x\sqrt{x^2-1}}\]
Work Step by Step
Let \[y=\csc^{-1} x\]
\[\Rightarrow x=\csc y\;\;\;...(1)\]
Differentiate (1) implicitly with respect to $x$
\[-\csc y\cot y\frac{dy}{dx}=1\]
\[\Rightarrow \frac{dy}{dx}=\frac{-1}{\csc y\cot y}\]
\[\Rightarrow \frac{dy}{dx}=\frac{-1}{\csc y\sqrt{\csc^2-1}}\]
Using (1)
\[\Rightarrow \frac{dy}{dx}=\frac{-1}{x\sqrt{x^2-1}}\]
Hence, \[\frac{dy}{dx}=\frac{-1}{x\sqrt{x^2-1}}\]
