1. The problem is to find the derivative of the function $y = e^{\frac{1}{x}}$. 2. We use the chain rule for differentiation: if $y = e^u$, then $y' = e^u \cdot u'$, where $u = \frac{1}{x}$. 3. Differentiate $u = \frac{1}{x}$: $$u' = -\frac{1}{x^2}$$ 4. Apply the chain rule: $$y' = e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) = -\frac{e^{\frac{1}{x}}}{x^2}$$ 5. So, the derivative of $y = e^{\frac{1}{x}}$ is: $$y' = -\frac{e^{\frac{1}{x}}}{x^2}$$