  We study the stability of the differential process of Rochberg and Weiss
associated to an analytic family of Banach spaces obtained using the complex
interpolation method for families. In the context of K\"othe function spaces we
complete earlier results of Kalton (who showed that there is global bounded
stability for pairs of K\"othe spaces) by showing that there is global
(bounded) stability for families of up to three K\"othe spaces distributed in
arcs on the unit sphere while there is no (bounded) stability for families of
four or more K\"othe spaces. In the context or arbitrary pairs of Banach spaces
we present local stability results and global isometric stability results.
