Answer
$$\frac{{dy}}{{dx}} = 2{e^{\sqrt 2 x}}$$
Work Step by Step
$$\eqalign{
& y = \sqrt 2 {e^{\sqrt 2 x}} \cr
& {\text{Find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\sqrt 2 {e^{\sqrt 2 x}}} \right] \cr
& \frac{{dy}}{{dx}} = \sqrt 2 \frac{d}{{dx}}\left[ {{e^{\sqrt 2 x}}} \right] \cr
& {\text{we can use the formula }}\frac{d}{{dx}}{e^u} = {e^u}\frac{{du}}{{dx}}\cr
&{\text{where }}u{\text{ is any differentiable function of }}x \cr
& {\text{note that }}u = \sqrt 2 x,{\text{ so}} \cr
& \frac{{dy}}{{dx}} = \sqrt 2 {e^{\sqrt 2 x}}\frac{d}{{dx}}\left[ {\sqrt 2 x} \right] \cr
& {\text{then}} \cr
& \frac{{dy}}{{dx}} = \sqrt 2 {e^{\sqrt 2 x}}\left( {\sqrt 2 } \right) \cr
& \frac{{dy}}{{dx}} = 2{e^{\sqrt 2 x}} \cr} $$
