Formal derivative

The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concept of a limit transition), and makes sense for any coefficient ring. For a polynomial

(or a power series ) the formal derivative is defined as (or , respectively), and for a rational function it is the rational function

Formal derivatives of higher order and formal partial derivatives of functions of several variables are defined similarly.

A number of properties of the ordinary derivative remain valid for the formal derivative. Thus, if , then is a constant in the coefficient field (in the case of characteristic 0) and is equal to (in the case of characteristic ). If is a root of multiplicity of a polynomial, then is a root of multiplicity of the derivative.