# 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.

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

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 ...
Formal Derivative - Correspondence To Analytic Derivative
... is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus ... the quotient (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 ...

### Famous quotes containing the words derivative and/or formal:

Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.
Henry David Thoreau (1817–1862)

It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.
Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)