  A torsion theory is called differential (higher differential) if a derivation
(higher derivation) can be extended from any module to the module of quotients
corresponding to the torsion theory. We study conditions equivalent to higher
differentiability of a torsion theory. It is known that the Lambek, Goldie and
any perfect torsion theories are differential. We show that these classes of
torsion theories are higher differential as well. Then, we study conditions
under which a higher derivation extended to a right module of quotients extends
also to a right module of quotients with respect to a larger torsion theory.
Lastly, we define and study the symmetric version of higher differential
torsion theories. We prove that the symmetric versions of the results on higher
differential (one-sided) torsion theories hold for higher derivations on
symmetric modules of quotients. In particular, we prove that the symmetric
Lambek, Goldie and any perfect torsion theories are higher differential.
