The differential of the exponential map on a symmetric space $M$ 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. 
Has anyone computed the same differential for the *tangent bundle* $TM$
of a symmetric space?