Answer
$$\frac{{dy}}{{dx}} = 0$$
Work Step by Step
$$\eqalign{
& y = {\sec ^{ - 1}}x + {\csc ^{ - 1}}x \cr
& {\text{Differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {{{\sec }^{ - 1}}x} \right) + \frac{d}{{dx}}\left( {{{\csc }^{ - 1}}x} \right) \cr
& {\text{Use the formulas }}\frac{d}{{dx}}\left[ {{{\sec }^{ - 1}}x} \right] = \frac{1}{{\left| x \right|\sqrt {{x^2} - 1} }}\,\,and\,\,\frac{d}{{dx}}\left[ {{{\csc }^{ - 1}}x} \right] = - \frac{1}{{\left| x \right|\sqrt {{x^2} - 1} }} \cr
& \frac{{dy}}{{dx}} = \frac{1}{{\left| x \right|\sqrt {{x^2} - 1} }} - \frac{1}{{\left| x \right|\sqrt {{x^2} - 1} }} \cr
& \frac{{dy}}{{dx}} = 0 \cr} $$
