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.

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.

Note that ${\displaystyle na_{n}}$ does not mean multiplication in the ring, but rather ${\displaystyle \sum _{k=1}^{n}a_{n},}$ where ${\displaystyle k}$ is never used inside the sum.

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