Answer
$$\frac{{dy}}{{dx}} = - 8x{e^{{x^2}}}$$
Work Step by Step
$$\eqalign{
& y = - 4{e^{{x^2}}} \cr
& {\text{differentiate both sides}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ { - 4{e^{{x^2}}}} \right] \cr
& \frac{{dy}}{{dx}} = - 4\frac{d}{{dx}}\left[ {{e^{{x^2}}}} \right] \cr
& {\text{use the rule }}\frac{d}{{dx}}\left[ {{e^{g\left( x \right)}}} \right] = {e^{g\left( x \right)}}g'\left( x \right) \cr
& \frac{{dy}}{{dx}} = - 4{e^{{x^2}}}\frac{d}{{dx}}\left[ {{x^2}} \right] \cr
& {\text{find derivative}} \cr
& \frac{{dy}}{{dx}} = - 4{e^{{x^2}}}\left( {2x} \right) \cr
& \frac{{dy}}{{dx}} = - 8x{e^{{x^2}}} \cr} $$