Derivative of Li 2(-1/x)

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Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{Li}_2 \left( -\dfrac{1}{x} \right) = \dfrac{\log \left(1+\frac{1}{x} \right)}{x} = \dfrac{\log(1+x)-\log(x)}{x},$$ where $\mathrm{Li}_2$ denotes the dilogarithm and $\log$ denotes the logarithm.

Proof

References

1981: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): $(1.6)$