The differential of the exponential map on a symmetric space can be expanded (abusing some notation) as

$d{\rm Exp}_X=\sum_{n=0}^{\infty}\frac{({\rm ad}X)^{2n}}{(2n+1)!}.$

This is an old (1958) result of Helgason.

Question (EDITED): Is there any generalization to reductive homogeneous spaces?

[EDIT 2017] I managed to solve the problem myself, so now it is a published
result. (A. V. Gavrilov, The Differential of the Exponential Map for Reductive Homogeneous Spaces, J. Lie Theory 25 (2015), No. 2, 363--376). The formula is more complicated then one might expect, so I omit it here.