Answer
$-\dfrac {e^{\frac {1}{x}}\left( 1+2x\right) }{x^{4}}$
Work Step by Step
$\dfrac {d}{dx}\left( \dfrac {e^{1/x}}{x^{2}}\right) =\dfrac {\dfrac {d}{dx}\left( e^{1/x}\right) {\times}x^{2}-\left( \dfrac {d}{dx}\left( x^{2}\right) \right) \times e^{1/x}}{x^{4}}=e^{1/x}\times x^{2}\times \dfrac {d}{dx}\left( \dfrac {1}{x}\right) -2x\times e^{\frac {1}{x}}=-\dfrac {e^{\frac {1}{x}}\left( 1+2x\right) }{x^{4}}$