Answer
$$\frac{{dy}}{{dx}} = {e^{ - 2x}}\left( {\cos x - 2\sin x} \right)$$
Work Step by Step
$$\eqalign{
& y = {e^{ - 2x}}\sin x \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = {D_x}\left( {{e^{ - 2x}}\sin x} \right) \cr
& {\text{use the product rule}} \cr
& \frac{{dy}}{{dx}} = {e^{ - 2x}} \cdot {D_x}\left( {\sin x} \right) + \sin x \cdot {D_x}\left( {{e^{ - 2x}}} \right) \cr
& {\text{solve derivatives}} \cr
& \frac{{dy}}{{dx}} = {e^{ - 2x}}\left( {\cos x} \right) + \sin x\left( { - 2{e^{ - 2x}}} \right) \cr
& {\text{multiply}} \cr
& \frac{{dy}}{{dx}} = {e^{ - 2x}}\cos x - 2{e^{ - 2x}}\sin x \cr
& {\text{factoring}} \cr
& \frac{{dy}}{{dx}} = {e^{ - 2x}}\left( {\cos x - 2\sin x} \right) \cr} $$
