Leibniz Rule

	Leibniz Rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, also called Leibniz's rule See also * Leibniz (other) * Leibniz' law (other) * List of things named after Gottfried Leibniz Gottfried Wilhelm Leibniz (1646–1716) was a German philosopher and mathematician. In engineering, the following concepts are attributed to Leibniz: * Leibniz wheel, a cylinder used in a class of mechanical calculators * Stepped reckoner, Leibni ... {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Product Rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negligi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	General Leibniz Rule In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if f and g are n-times differentiable functions, then the product fg is also n-times differentiable and its nth derivative is given by :(fg)^=\sum_^n f^ g^, where = is the binomial coefficient and f^ denotes the ''j''th derivative of ''f'' (and in particular f^= f). The rule can be proved by using the product rule and mathematical induction. Second derivative If, for example, , the rule gives an expression for the second derivative of a product of two functions: :(fg)''(x)=\sum\limits_^=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x). More than two factors The formula can be generalized to the product of ''m'' differentiable functions ''f''1,...,''f''''m''. :\left(f_1 f_2 \cdots f_m\right)^=\sum_ \prod_f_^\,, where the sum extends over all ''m''-tuples (''k''1,...,''k''''m'') of non-negative integers with \sum_^m k_t=n, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Leibniz Integral Rule In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are functions dependent on $x,$ the derivative of this integral is expressible as $\frac \left (\int_^ f(x,t)\,dt \right )= f\big(x,b(x)\big)\cdot \frac b(x) - f\big(x,a(x)\big)\cdot \frac a(x) + \int_^\frac f(x,t) \,dt,$ where the partial derivative $\tfrac$ indicates that inside the integral, only the variation of $f(x, t)$ with $x$ is considered in taking the derivative. In the special case where the functions $a(x)$ and $b(x)$ are constants $a( ...$ [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Alternating Series Test In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. Formal Statement Alternating series test A series of the form : \sum_^\infty (-1)^ a_n = a_0-a_1 + a_2 - a_3 + \cdots \! where either all ''a''''n'' are positive or all ''a''''n'' are negative, is called an alternating series. The alternating series test guarantees that an alternating series converges if the following two conditions are met: # , a_n, decreases monotonically, i.e., , a_, \leq, a_n, , and # \lim_ a_n = 0 Alternating series estimation theorem Moreover, let ''L'' denote the sum of the series, then the partial s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Leibniz (other) Gottfried Wilhelm Leibniz (1646–1716) was a German philosopher and mathematician. Leibniz may also refer to: Friedrich Leibniz (1597–1652), father of Gottfried Leibniz Leibniz or Leibniz-Keks The Leibniz-Keks or Choco Leibniz is a German brand of biscuit or cookie produced by the Bahlsen food company since 1891. It was created by the firm as a rival to a similar French biscuit, the Petit-Beurre. Name The brand name ''Leibniz'' come ..., a brand of biscuit See also * Leibnitz (other) * List of things named after Gottfried Leibniz {{Disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Leibniz' Law (other) Leibniz' law may refer to: * The product rule * General Leibniz rule, a generalization of the product rule * Identity of indiscernibles See also Leibniz (other) Gottfried Wilhelm Leibniz (1646–1716) was a German philosopher and mathematician. Leibniz may also refer to: Friedrich Leibniz (1597–1652), father of Gottfried Leibniz Leibniz or Leibniz-Keks The Leibniz-Keks or Choco Leibniz is a German b ... Leibniz's rule (other) {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]