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. 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 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: Properties 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 When R is not commutative there is another, different, linearity property in which r and s appear on the right rather than on the left. When R does not contain an identity element, neither of these reduces to the case of simply a sum of polynomials or the sum of a polynomial with a multiple of another polynomial, which must also be included as a "linearity"