The $\mathcal{L}$-invariant, the dual $\mathcal{L}$-invariant, and families

Given a rank two trianguline family of $(\phi ,\mathrm{\Gamma })$-modules having a noncrystalline semistable member, we compute the Fontaine–Mazur $\mathcal{L}$-invariant of that member in terms of the logarithmic derivative, with respect to the Sen weight, of the value at p of the trianguline parameter. This generalizes prior work, in the case of Galois representations, due to Greenberg–Stevens and Colmez.