Hasse derivative

In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties.
Definition.
Let "k"["X"] be a polynomial ring over a field "k". The "r"-th Hasse derivative of "X""n" is
if "n" ≥ "r" and zero otherwise. In characteristic zero we have
Properties.
The Hasse derivative is a generalized derivation on "k"["X"] and extends to a generalized derivation on the function field "k"("X"), satisfying an analogue of the product rule 
and an analogue of the chain rule. Note that the formula_4 are not themselves derivations in general, but are closely related.
A form of Taylor's theorem holds for a function "f" defined in terms of a local parameter "t" on an algebraic variety: