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. 



The differential of the exponential map: reductive homogeneous space
I think the formula will be essentially the same. 


The formula you wrote is valid in general for the exponential map of analytic manifolds equipped with an analytic affine connection. It is stated and proved in [this paper](http://www.mscand.dk/article.php?id=1601) by Helgason (see pages 6-7 of the linked .pdf): *Some remarks on the exponential mapping of an affine connection. Math. Scand. 9 (l961), l29-l46.*


The existence of such connections on reductive homogeneous spaces, with an additional invariance property (that you might probably need at some point) is stated and proved in *K.Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65*. 


