  We give an explicit presentation of the module of differentials of order $n$
of a finitely generated algebra via a higher-order Jacobian matrix. We use the
presentation to study some aspects of this module in the case of hypersurfaces.
More precisely, we prove higher-order versions of known results relating
freness and torsion-freness of the module of differentials with the regularity
and normality of the hypersurface. We also study its projective dimension.
