In
mathematics, the formal derivative is an operation on elements of a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
or a ring of
formal power series that mimics the form of the derivative from
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, 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.
Definition
The definition of formal derivative is as follows: fix a ring ''R'' (not necessarily commutative) and let ''A'' = ''R''
'x''be the ring of polynomials over ''R''. Then the formal derivative is an operation on elements of ''A'', where if
:
then its formal derivative is
:
just as for polynomials over the
real or
complex numbers. Here
does not mean multiplication in the ring, but rather
where
is never used inside the sum.
There is a problem with this definition for noncommutative rings. The formula itself is correct, but there is no standard form of a polynomial. Therefore using this definition it is difficult to prove that
Axiomatic definition well suited for noncommutative rings
As opposed to the above formula one may define the formal derivative axiomatically as the map