Formal derivative

In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general impossible to define for a ring. Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not. Formal differentiation is used in algebra to test for multiple roots of a polynomial. Fix a ring (not necessarily commutative) and let be the ring of polynomials over . (If is not commutative, this is the Free algebra over a single indeterminate variable.) Then the formal derivative is an operation on elements of , where if then its formal derivative is In the above definition, for any nonnegative integer and , is defined as usual in a Ring: (with if ). This definition also works even if does not have a multiplicative identity. One may also define the formal derivative axiomatically as the map satisfying the following properties.

1. for all
2. The normalization axiom,
3. The map commutes with the addition operation in the polynomial ring,
4. The map satisfies Leibniz's law with respect to the polynomial ring's multiplication operation, One may prove that this axiomatic definition yields a well-defined map respecting all of the usual ring axioms. The formula above (i.e. the definition of the formal derivative when the coefficient ring is commutative) is a direct consequence of the aforementioned axioms: It can be verified that: Formal differentiation is linear: for any two polynomials f(x),g(x) in R[x] and elements r,s of R we have The formal derivative satisfies the Product rule: Note the order of the factors; when R is not commutative this is important. These two properties make D a derivation on A (see module of relative differential forms for a discussion of a generalization).