Differential of arcsec x
To find\[\frac{d}{dx}(sec^{-1}x)\]
;et \[y=sec^{-1}x\]
then \[secy=x\]
.Differentiating implicitly gives
\[secytany \frac{dy}{dx}=1 \rightarrow \frac{dy}{dx}= \frac{1}{secy tany}\]
.To express
\[\frac{dy}{dx}\]
in terms of \[x\]
use \[secy=x,\; tany = \sqrt{sec^2y-1} =\sqrt{x^2-1}\]
.Then
\[\frac{dy}{dx}= \frac{1}{x \sqrt{x^2-1}}\]
.