Answer
$$\frac{{dy}}{{dx}} = - 6{x^2}{e^{ - 2{x^3}}}$$
Work Step by Step
$$\eqalign{
& y = {e^{ - 2{x^3}}} \cr
& {\text{differentiate both sides}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{e^{ - 2{x^3}}}} \right] \cr
& {\text{use the rule }}\frac{d}{{dx}}\left[ {{e^{g\left( x \right)}}} \right] = {e^{g\left( x \right)}}g'\left( x \right) \cr
& \frac{{dy}}{{dx}} = {e^{ - 2{x^3}}}\frac{d}{{dx}}\left[ { - 2{x^3}} \right] \cr
& {\text{find derivative}} \cr
& \frac{{dy}}{{dx}} = {e^{ - 2{x^3}}}\left( { - 6{x^2}} \right) \cr
& \frac{{dy}}{{dx}} = - 6{x^2}{e^{ - 2{x^3}}} \cr} $$
