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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 linearity of differentiation ： ウィキペディア英語版 linearity of differentiation In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions;〔.〕 this property is known as linearity of differentiation, the rule of linearity,〔.〕 or the superposition rule for differentiation.〔.〕 It is a fundamental property of the derivative that encapsulates in a single rule two simpler rules of differentiation, the sum rule (the derivative of the sum of two functions is the sum of the derivatives) and the constant factor rule (the derivative of a constant multiple of a function is the same constant multiple of the derivative).〔.〕〔.〕 Thus it can be said that the act of differentiation is linear, or the differential operator is a linear operator.〔.〕 ==Statement and derivation== Let and be functions, with and constants. Now consider: :$\backslash frac\; x\}\; (\; \backslash alpha\; \backslash cdot\; f(x)\; +\; \backslash beta\; \backslash cdot\; g(x)\; )$ By the sum rule in differentiation, this is: :$\backslash frac\; x\}\; (\; \backslash alpha\; \backslash cdot\; f(x)\; )\; +\; \backslash frac\; x\}\; (\backslash beta\; \backslash cdot\; g(x))$ By the constant factor rule in differentiation, this reduces to: :$\backslash alpha\; \backslash cdot\; f\text{'}(x)\; +\; \backslash beta\; \backslash cdot\; g\text{'}(x)$ This in turn leads to: :$\backslash frac\; x\}(\backslash alpha\; \backslash cdot\; f(x)\; +\; \backslash beta\; \backslash cdot\; g(x))\; =\; \backslash alpha\; \backslash cdot\; f\text{'}(x)\; +\; \backslash beta\; \backslash cdot\; g\text{'}(x)$ Omitting the brackets, this is often written as: :$(\backslash alpha\; \backslash cdot\; f\; +\; \backslash beta\; \backslash cdot\; g)\text{'}\; =\; \backslash alpha\; \backslash cdot\; f\text{'}+\; \backslash beta\; \backslash cdot\; g\text{'}$ 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「linearity of 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.