  In this paper we study the derivatives of Frobenius and the derivatives of
Hodge weights for families of Galois representations with triangulations. We
generalize the Fontaine-Mazur L-invariant and use it to build a formula which
is a generalization of the Greenberg-Stevens-Colmez formula. For the purpose of
proving this formula we show two auxiliary results called projection vanishing
property and "projection vanishing implying L-invariants" property.
