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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 Leibniz integral rule ： ウィキペディア英語版 Leibniz integral rule In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form : $\backslash int\_^\; f(x,\; y)\; \backslash ,\backslash mathrmy$ then for ''x'' in (''x''0, ''x''1) the derivative of this integral is thus expressible : $x\}\; \backslash left\; (\; \backslash int\_^\; f(x,\; y)\; \backslash ,\backslash mathrmy\; \backslash right\; )=\; \backslash int\_^\; f\_x(x,y)\backslash ,\backslash mathrmy$ provided that ''f'' and its partial derivative'' fx'' are both continuous over a region in the form (''x''1 ) × (''y''1 ). Thus under certain conditions, one may interchange the integral and partial differential operators. This important result is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits. == Formal statement == Let ''f''(''x'', ''t'') be a function such that the partial derivative of ''f'' with respect to ''t'' exists, and is continuous. Then, :$\backslash fract\}\; \backslash left\; (\backslash int\_^\; f(x,t)\backslash ,\backslash mathrmx\; \backslash right\; )=\; \backslash int\_^\backslash frac\backslash ,\backslash mathrmx\; \backslash ,+\backslash ,\; f\backslash big(b(t),t\backslash big)\backslash cdot\; b\text{'}(t)\; \backslash ,-\backslash ,\; f\backslash big(a(t),t\; \backslash big)\backslash cdot\; a\text{'}(t)$ where the partial derivative indicates that inside the integral, only the variation of ''f''(•,''t'') with ''t'' is considered in taking the derivative. 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「Leibniz integral rule」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.