Answer
$$\frac{{dy}}{{dx}} = - \frac{1}{{{{\cos }^{ - 1}}x\sqrt {1 - {x^2}} }}$$
Work Step by Step
$$\eqalign{
& y = \ln {\cos ^{ - 1}}x \cr
& {\text{find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{{d\left( {\ln \left( {{{\cos }^{ - 1}}x} \right)} \right)}}{{dx}} \cr
& {\text{ use the formula }}\cr
&\frac{{d\left( {\ln u} \right)}}{{dx}} = \frac{1}{u}\frac{{du}}{{dt}}.\,\,\,\left( {{\text{see table 7}}{\text{.3}}} \right). \cr
& \frac{{dy}}{{dx}} = \frac{1}{{{{\cos }^{ - 1}}x}}\frac{{d\left( {{{\cos }^{ - 1}}x} \right)}}{{dx}} \cr
& {\text{we can use the formula }}\cr
&\frac{{d\left( {{{\cos }^{ - 1}}x} \right)}}{{dx}} = - \frac{1}{{\sqrt {1 - {x^2}} }}\frac{{du}}{{dx}}.\,\,\,\left( {{\text{see table 7}}{\text{.3}}} \right). \cr
& {\text{here }}u = x,\,\,{\text{so}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{{{\cos }^{ - 1}}x}}\left( { - \frac{1}{{\sqrt {1 - {x^2}} }}} \right)\frac{{d\left( x \right)}}{{dx}} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{{{\tan }^{ - 1}}x}}\left( { - \frac{1}{{\sqrt {1 - {x^2}} }}} \right)\left( 1 \right) \cr
& {\text{simplifying, we get:}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{{{\cos }^{ - 1}}x\sqrt {1 - {x^2}} }} \cr} $$
