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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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408101.png" /></td> </tr></table>
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$$
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F ( X)  = \
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\sum _ {i = 0 } ^ { n }
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a _ {i} X  ^ {i}
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$$
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(or a power series  $  A ( X) = \sum _ {i = 0 }  ^  \infty  b _ {i} X  ^ {i} $)
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the formal derivative  $  F ^ { \prime } ( X) $
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is defined as  $  \sum _ {i = 1 }  ^ {n} ia _ {i} X ^ {i - 1 } $(
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or  $  \sum _ {i = 1 }  ^  \infty  ib _ {i} X ^ {i - 1 } $,
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respectively), and for a rational function  $  f ( X) = P ( X)/Q ( X) $
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it is the rational function
  
(or a power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408102.png" />) the formal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408103.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408104.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408105.png" />, respectively), and for a rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408106.png" /> it is the rational function
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$$
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f ^ { \prime } ( X) = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408107.png" /></td> </tr></table>
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\frac{P ^ { \prime } ( X) Q ( X) - Q  ^  \prime  ( X) P ( X) }{[ Q ( X)]  ^ {2} }
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.
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$$
  
 
Formal derivatives of higher order and formal partial derivatives of functions of several variables are defined similarly.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408108.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f0408109.png" /> is a constant in the coefficient field (in the case of characteristic 0) and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f04081010.png" /> (in the case of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f04081011.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f04081012.png" /> is a root of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f04081013.png" /> of a polynomial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f04081014.png" /> is a root of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040810/f04081015.png" /> of the derivative.
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A number of properties of the ordinary derivative remain valid for the formal derivative. Thus, if $  F ^ { \prime } ( X) = 0 $,  
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then $  F ( X) $
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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.

Latest revision as of 19:39, 5 June 2020


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