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Formal derivative

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The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concept of a limit transition), and makes sense for any coefficient ring. For a polynomial

$$ F ( X) = \ \sum _ {i = 0 } ^ { n } a _ {i} X ^ {i} $$

(or a power series $ A ( X) = \sum _ {i = 0 } ^ \infty b _ {i} X ^ {i} $) the formal derivative $ F ^ { \prime } ( X) $ is defined as $ \sum _ {i = 1 } ^ {n} ia _ {i} X ^ {i - 1 } $( or $ \sum _ {i = 1 } ^ \infty ib _ {i} X ^ {i - 1 } $, respectively), and for a rational function $ f ( X) = P ( X)/Q ( X) $ it is the rational function

$$ f ^ { \prime } ( X) = \ \frac{P ^ { \prime } ( X) Q ( X) - Q ^ \prime ( X) P ( X) }{[ Q ( X)] ^ {2} } . $$

Formal derivatives of higher order and formal partial derivatives of functions of several variables are defined similarly.

A number of properties of the ordinary derivative remain valid for the formal derivative. Thus, if $ F ^ { \prime } ( X) = 0 $, then $ F ( X) $ is a constant in the coefficient field (in the case of characteristic 0) and is equal to $ G ( X ^ {p} ) $( in the case of characteristic $ p $). If $ x _ {0} $ is a root of multiplicity $ k $ of a polynomial, then $ x _ {0} $ is a root of multiplicity $ k - 1 $ of the derivative.

How to Cite This Entry:
Formal derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Formal_derivative&oldid=46954
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article