Faddeev's noncompact quantum dilogarithm is the function defined by$$ \Phi_{\mathsf b}(z) = \exp \int_{\mathbb{R} + i\varepsilon} \frac{ e^{-2i zw} }{ 4 \sinh(w \mathsf b ) \sinh(w/\mathsf b) } \frac{dw}{w}$$for $|\Im z| < |\Im c_{\mathsf b}|$, where $c_{\mathsf b} = \frac{i}{2} \left( \mathsf b + \mathsf b^{-1}\right)$. It can be analytically continued to a meromorphic function on $\mathbb{C}$ with a number of interesting properties. (For example, see [1, Appendix A].) In general, it is a sort of quantum analogue of the (exponentiated) dilogarithm function$L(z) = e^{\operatorname{Li}_2(e^z)/2\pi i}$where$\operatorname{Li}_2(z)=-\int_{0}^{z} \frac{\log(1-t)}{t} dt$is the usual dilogarithm.
Has anyone studied the derivative $\Phi_{\mathsf b}'(z)$, or the logarithmic derivative $\Phi_{\mathsf b}'(z)/\Phi_{\mathsf b}(z)$? Presumably it would be an analogue of the logarithmic derivative $- \log(1 - e^z)$ of $L(z)$ (here I'm ignoring some factors of $2\pi i$).
[1] "A TQFT from quantum Teichmueller Theory" arXiv:1109.6295