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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 differentiation ： ウィキペディア英語版 Logarithmic differentiation In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function ''f'', :$()\text{'}\; =\; \backslash frac\; \backslash quad\; \backslash rightarrow\; \backslash quad\; f\text{'}\; =\; f\; \backslash cdot\; ()\text{'}.$ The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base ''e'') to transform products into sums and divisions into subtractions. The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero. ==Overview== For a function :$y=f(x)\backslash ,\backslash !$ logarithmic differentiation typically begins by taking the natural logarithm, or the logarithm to the base ''e'', on both sides, remembering to take absolute values :$\backslash ln|y|\; =\; \backslash ln|f(x)|\backslash ,\backslash !$ After implicit differentiation :$\backslash frac\; \backslash frac\; =\; \backslash frac$ Multiplication by ''y'' is then done to eliminate 1/''y'' and leave only ''dy''/''dx'' on the left-hand side: :$\backslash frac\; =\; y\; \backslash times\; \backslash frac\; =\; f\text{'}(x).$ The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated. These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws:〔 : $\backslash ln(ab)\; =\; \backslash ln(a)\; +\; \backslash ln(b),\; \backslash qquad$ \ln\left(\frac\right) = \ln(a) - \ln(b), \qquad \ln(a^n) = n\ln(a) 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 differentiation」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.