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?