|
Fundamental Increment Lemma
In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative f'(a) of a function f at a point a: :f'(a) = \lim_ \frac. The lemma asserts that the existence of this derivative implies the existence of a function \varphi such that :\lim_ \varphi(h) = 0 \qquad \text \qquad f(a+h) = f(a) + f'(a)h + \varphi(h)h for sufficiently small but non-zero h. For a proof, it suffices to define :\varphi(h) = \frac - f'(a) and verify this \varphi meets the requirements. The lemma says, at least when h is sufficiently close to zero, that the difference quotient :\frac can be written as the derivative ''f plus an error term \varphi(h) that vanishes at h=0. That is, one has :\frac = f'(a) + \varphi(h). Differentiability in higher dimensions In that the existence of \varphi uniquely characterises the number f'(a), the fundamental increment lemma can be said to characterise the differentiability of single-variable fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Differential Calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a Function (mathematics), function, related notions such as the Differential of a function, differential, and their applications. The derivative of a function at a chosen input value describes the Rate (mathematics)#Of_change, rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent, tangent line to the graph of a function, graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
