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 derivative, derivative, formal":
... 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 ...
... 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 ...
Famous quotes containing the words derivative and/or formal:
“When we say science we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science the vulgarized derivative from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.”
—Wyndham Lewis (18821957)
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—William Shakespeare (15641616)