Answer
$$\frac{{dy}}{{dx}} = - 2{x^{ - \sqrt 2 - 1}}$$
Work Step by Step
$$\eqalign{
& y = \sqrt 2 {x^{ - \sqrt 2 }} \cr
& {\text{Find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\sqrt 2 {x^{ - \sqrt 2 }}} \right] \cr
& \frac{{dy}}{{dx}} = \sqrt 2 \frac{d}{{dx}}\left[ {{x^{ - \sqrt 2 }}} \right] \cr
& {\text{use the power rule: }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr
& \frac{{dy}}{{dx}} = \sqrt 2 \left( { - \sqrt 2 {x^{ - \sqrt 2 - 1}}} \right) \cr
& \frac{{dy}}{{dx}} = - 2{x^{ - \sqrt 2 - 1}} \cr} $$
