First simplify the power: $\displaystyle \frac{1}{x}\left(x-\frac{1}{x}\right)=1-\frac{1}{x^2}$ , and now apply the chain rule to $\displaystyle \left(e^{f(x)}\right)'=f'(x)\cdot e^{f(x)}$ ...
Do you see why Tonio was confused? (Because he never gives a wrong answer!)
What you gave in your initial post was $\displaystyle e^{\frac{1}{x}\left(x- \frac{1}{x}\right)}$ not $\displaystyle e^{\frac{1}{x}}\left(x- \frac{1}{x}\right)$.