Answer
$$\frac{{dy}}{{dx}} = x{e^x}$$
Work Step by Step
$$\eqalign{
& y = x{e^x} - {e^x} \cr
& {\text{Find the derivative of }}y{\text{ with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {x{e^x} - {e^x}} \right] \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {x{e^x}} \right] - \frac{d}{{dx}}\left[ {{e^x}} \right] \cr
& {\text{use the product rule for }}\frac{d}{{dx}}\left[ {x{e^x}} \right] \cr
& \frac{{dy}}{{dx}} = x\frac{d}{{dx}}\left[ {{e^x}} \right] + {e^x}\frac{d}{{dx}}\left[ x \right] - \frac{d}{{dx}}\left[ {{e^x}} \right] \cr
& {\text{solve the derivatives and simplify}} \cr
& \frac{{dy}}{{dx}} = x\left( {{e^x}} \right) + {e^x}\left( 1 \right) - {e^x} \cr
& \frac{{dy}}{{dx}} = x{e^x} + {e^x} - {e^x} \cr
& \frac{{dy}}{{dx}} = x{e^x} \cr} $$
