If \(f(x)=e^{1/x}\), then \(f'(x)=\)

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We will combine the power rule and exponential rule for \(e^x\) as shown:

$$ \frac{d}{dx}e^x = e^x \tag*{\tiny exponential rule for parent function \(e^x\)} $$ $$ \frac{d}{dx}\left(\frac{1}{x}\right)=\frac{d}{dx}(x^{-1}) = -1\cdot x^{-2} = -\frac{1}{x^2} \tag*{\tiny power rule} $$

Chain rule:

$$\frac{d}{dx}e^{1/x}=e^{1/x}\cdot -\frac{1}{x^2} $$ $$=\boxed{-\frac{e^{1/x}}{x^2}}$$