Differentiatial of Arccosec x

To find  
\[\frac{d}{dx}(cosec^{-1}x)\]
  let  
\[y=cosec^{-1}x\]
  then  
\[cosecy=x\]
.
Differentiating implicitly gives  
\[-cosecycoty \frac{dy}{dx}=1 \rightarrow \frac{dy}{dx}=- \frac{1}{cosecy coty}\]
.
To express  
\[\frac{dy}{dx}\]
  in terms of  
\[x\]
  use  
\[cosecy=x,\; coty = \sqrt{cosec^2y-1} =\sqrt{x^2-1}\]
.
Then  
\[\frac{dy}{dx}=- \frac{1}{x \sqrt{x^2-1}}\]
.