翻訳と辞書 Words near each other ・ Inverse Cost and Quality Law ・ Inverse curve ・ Inverse demand function ・ Inverse distance weighting ・ Inverse distribution ・ Inverse dynamics ・ Inverse electron-demand Diels–Alder reaction ・ Inverse element ・ Inverse exchange-traded fund ・ Inverse Faraday effect ・ Inverse filter ・ Inverse filter (disambiguation) ・ Inverse floating rate note ・ Inverse function ・ Inverse function theorem ・ Inverse functions and differentiation ・ Inverse Galois problem ・ Inverse gambler's fallacy ・ Inverse gas chromatography ・ Inverse Gaussian distribution ・ Inverse hyperbolic function ・ Inverse image functor ・ Inverse iteration ・ Inverse kinematics ・ Inverse Laplace transform ・ Inverse limit ・ Inverse magnetostrictive effect ・ Inverse mapping theorem ・ Inverse matrix gamma distribution ・ Inverse mean curvature flow Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Inverse functions and differentiation ： ウィキペディア英語版 Inverse functions and differentiation In mathematics, the inverse of a function $y\; =\; f(x)$ is a function that, in some fashion, "undoes" the effect of $f$ (see inverse function for a formal and detailed definition). The inverse of $f$ is denoted $f^$. The statements and are equivalent. Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is: :$\backslash frac\backslash ,\backslash cdot\backslash ,\; \backslash frac\; =\; 1.$ This is a direct consequence of the chain rule, since :$\backslash frac\backslash ,\backslash cdot\backslash ,\; \backslash frac\; =\; \backslash frac$ and the derivative of $x$ with respect to $x$ is 1. Writing explicitly the dependence of $y$ on $x$ and the point at which the differentiation takes place and using Lagrange's notation, the formula for the derivative of the inverse becomes :$\backslash left()\text{'}(a)=\backslash frac$ Geometrically, a function and inverse function have graphs that are reflections, in the line . This reflection operation turns the gradient of any line into its reciprocal. Assuming that $f$ has an inverse in a neighbourhood of $x$ and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at $x$ and have a derivative given by the above formula. ==Examples== * $\backslash ,y\; =\; x^2$ (for positive $x$) has inverse $x\; =\; \backslash sqrt$. :$\backslash frac\; =\; 2x$ \mbox\mbox\mbox\mbox; \mbox\mbox\mbox\mbox \frac = \frac :$\backslash frac\backslash ,\backslash cdot\backslash ,\backslash frac\; =\; 2x\; \backslash cdot\backslash frac\; =\; 1.$ At , however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function. * $\backslash ,y\; =\; e^x$ (for real $x$) has inverse $\backslash ,x\; =\; \backslash ln$ (for positive $y$) :$\backslash frac\; =\; e^x$ \mbox\mbox\mbox\mbox; \mbox\mbox\mbox\mbox \frac = \frac :$\backslash frac\backslash ,\backslash cdot\backslash ,\backslash frac\; =\; e^x\; \backslash cdot\; \backslash frac\; =\; \backslash frac\; =\; 1$ 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「Inverse functions and 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.