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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Formal derivative ： ウィキペディア英語版 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. ==Definition== The definition of formal derivative is as follows: fix a ring ''R'' (not necessarily commutative) and let ''A'' = ''R''() be the ring of polynomials over ''R''. Then the formal derivative is an operation on elements of ''A'', where if :$f(x)\backslash ,=\backslash ,a\_n\; x^n\; +\; \backslash cdots\; +\; a\_1\; x\; +\; a\_0$ then its formal derivative is :$f\text{'}(x)\backslash ,=\backslash ,Df(x)\; =\; n\; a\_n\; x^\; +\; \backslash cdots\; +\; 2\; a\_2\; x\; +\; a\_1$ just as for polynomials over the real or complex numbers. Note that $n\; a\_n$ does not mean multiplication in the ring, but rather $\backslash sum\_^n\; a\_n,$ where $k$ is never used inside the sum. One should mention that 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's difficult to prove $(f(x)\backslash cdot\; b)\text{'}=f\text{'}(x)\backslash cdot\; b$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Formal derivative」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.