Answer
$$\frac{{dy}}{{dx}} = - \frac{{2x}}{{\sqrt {1 - {x^4}} }}$$
Work Step by Step
$$\eqalign{
& y = {\cos ^{ - 1}}\left( {{x^2}} \right) \cr
& {\text{find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{{d\left( {{{\cos }^{ - 1}}\left( {{x^2}} \right)} \right)}}{{dx}} \cr
& {\text{we can use the formula }}\frac{{d\left( {{{\cos }^{ - 1}}u} \right)}}{{dx}} = - \frac{1}{{\sqrt {1 - {u^2}} }}\frac{{du}}{{dx}}.\,\,\,\left( {{\text{see table 7}}{\text{.3}}} \right). \cr
& {\text{here }}u = {x^2},\,\,{\text{then}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{\sqrt {1 - {{\left( {{x^2}} \right)}^2}} }}\frac{{d\left( {{x^2}} \right)}}{{dx}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{\sqrt {1 - {x^4}} }}\left( {2x} \right) \cr
& \frac{{dy}}{{dx}} = - \frac{{2x}}{{\sqrt {1 - {x^4}} }} \cr} $$
