Answer
$(2x+1)e^{-1/x}$
Work Step by Step
$\frac{d}{dx}x^2e^{-1/x}$
Use chain rule:
$=x^2\frac{d}{dx}e^{-1/x}+e^{-1/x}\frac{d}{dx}x^2$
Use chain rule again to evaluate $\frac{d}{dx}e^{-1/x}$:
$=x^2e^{-1/x}\frac{d}{dx}(-\frac{1}{x})+e^{-1/x}*2x$
$=x^2e^{-1/x}\frac{d}{dx}(-x^{-1})+e^{-1/x}*2x$
$=x^2e^{-1/x}x^{-2}+e^{-1/x}*2x$
$=e^{-1/x}+e^{-1/x}*2x$
$=(2x+1)e^{-1/x}$