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In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by : where is the derivative of ''f'' with respect to ''x'', and ''dx'' is an additional real variable (so that ''dy'' is a function of ''x'' and ''dx''). The notation is such that the equation : holds, where the derivative is represented in the Leibniz notation ''dy''/''dx'', and this is consistent with regarding the derivative as the quotient of the differentials. One also writes : The precise meaning of the variables ''dy'' and ''dx'' depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables ''dx'' and ''dy'' are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis. ==History and usage== The differential was first introduced via an intuitive or heuristic definition by Gottfried Wilhelm Leibniz, who thought of the differential ''dy'' as an infinitely small (or infinitesimal) change in the value ''y'' of the function, corresponding to an infinitely small change ''dx'' in the function's argument ''x''. For that reason, the instantaneous rate of change of ''y'' with respect to ''x'', which is the value of the derivative of the function, is denoted by the fraction : in what is called the Leibniz notation for derivatives. The quotient ''dy''/''dx'' is not infinitely small; rather it is a real number. The use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy (1823) defined the differential without appeal to the atomism of Leibniz's infinitesimals.〔For a detailed historical account of the differential, see , especially page 275 for Cauchy's contribution on the subject. An abbreviated account appears in .〕〔Cauchy explicitly denied the possibility of actual infinitesimal and infinite quantities , and took the radically different point of view that "a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge to zero" (; translation from ).〕 Instead, Cauchy, following d'Alembert, inverted the logical order of Leibniz and his successors: the derivative itself became the fundamental object, defined as a limit of difference quotients, and the differentials were then defined in terms of it. That is, one was free to ''define'' the differential ''dy'' by an expression : in which ''dy'' and ''dx'' are simply new variables taking finite real values, not fixed infinitesimals as they had been for Leibniz.〔: "The differentials as thus defined are only new ''variables'', and not fixed infinitesimals..."〕 According to , Cauchy's approach was a significant logical improvement over the infinitesimal approach of Leibniz because, instead of invoking the metaphysical notion of infinitesimals, the quantities ''dy'' and ''dx'' could now be manipulated in exactly the same manner as any other real quantities in a meaningful way. Cauchy's overall conceptual approach to differentials remains the standard one in modern analytical treatments,〔: "Here we remark merely in passing that it is possible to use this approximate representation of the increment Δ''y'' by the linear expression ''hƒ''(''x'') to construct a logically satisfactory definition of a "differential", as was done by Cauchy in particular."〕 although the final word on rigor, a fully modern notion of the limit, was ultimately due to Karl Weierstrass. In physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. reconcile the physical use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the particular purpose for which they are intended. Thus "physical infinitesimals" need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense. Following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real analysis, it is more desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a linear functional of an increment Δ''x''. This approach allows the differential (as a linear map) to be developed for a variety of more sophisticated spaces, ultimately giving rise to such notions as the Fréchet or Gâteaux derivative. Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: the exterior derivative of the function. In non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing (see differential (infinitesimal)). 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「differential of a function」の詳細全文を読む スポンサード リンク
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