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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 Logarithmic derivative ： ウィキペディア英語版 Logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula : $\backslash frac\; \backslash !$ where $f\text{'}$ is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f''; that is, the infinitesimal absolute change in ''f,'' namely $f\text{'},$ scaled by the current value of ''f.'' When ''f'' is a function ''f''(''x'') of a real variable ''x'', and takes real, strictly positive values, this is equal to the derivative of ln(''f''), or the natural logarithm of ''f''. This follows directly from the chain rule. ==Basic properties== Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have : $(\backslash log\; uv)\text{'}\; =\; (\backslash log\; u\; +\; \backslash log\; v)\text{'}\; =\; (\backslash log\; u)\text{'}\; +\; (\backslash log\; v)\text{'}\; .\backslash !$ So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get : $\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; .\backslash !$ Thus, it is true for ''any'' function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined). A corollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function: : $\backslash frac\; =\; \backslash frac\; =\; -\backslash frac\; ,\backslash !$ just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number. More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor: : $\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; -\; \backslash frac\; ,\backslash !$ just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor. Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base: : $\backslash fracu\text{'}\}\; ,\backslash !$ just as the logarithm of a power is the product of the exponent and the logarithm of the base. In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic 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「Logarithmic 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.