Derivatives of Frobenius and Derivatives of Hodge–Tate Weights

In this paper we study the derivatives of Frobenius and the derivatives of Hodge–Tate 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 Colmez–Greenberg–Stevens formula.For the purpose of proving this formula we show two auxiliary results called projection vanishing property and"projection vanishing implying L-invariants"property.