Let \(\displaystyle f(x) = \frac 1{1+e^{1/x}}\). If

\[PV\ \int_{-1}^{1} \frac{d}{dx} f(x) \ dx = \lambda + \frac{1-e}{1+e},\]

where \(\lambda\) is some constant, what is the value of \(\lambda\)?

**Note**: The "\(PV\)" before the integral indicates the Cauchy principal value.

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