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. The primary use of formal differentiation in algebra is to test for multiple roots of a polynomial.

Read more about Formal DerivativeDefinition, Properties, Application To Finding Repeated Factors, Correspondence To Analytic Derivative

Other articles related to "formal derivative, derivative, formal":

Formal Derivative - Correspondence To Analytic Derivative
... commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus ... (in R) by g then it is not hard to verify that g(X,X) (in R) coincides with the formal derivative of f as it was defined above ... This formulation of the derivative works equally well for a formal power series, assuming only that the ring of scalars is commutative ...
Operations On Formal Power Series - Formal Differentiation of Series
... Given a formal power series in R], we define its formal derivative, denoted Df or, by The symbol D is called the formal differentiation operator ... Additionally, the formal derivative has many of the properties of the usual derivative of calculus ... Thus, in these respects formal power series behave like Taylor series ...

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