Leibniz's Rule (derivatives And Integrals), Leibniz's 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 * Leibniz calculator, a ...

Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. 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. Leibniz contributed to the field ...
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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 "infinitesimals" (a precursor to the modern differential). (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'' and ''v'' be functions. Then ''d(uv)'' is the same thing as the difference between two successive ''uvs; let one of these be ''uv'', and the other ''u+du'' times ''v+dv''; then: \begin d(u\cdot v) & = (u + d ...
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General Leibniz Rule
In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule for the derivative of the product of two (which is also known as "Leibniz's rule"). It states that if f and g are -times differentiable functions, then the product fg is also -times differentiable and its -th 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 proven 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 ...
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Leibniz Integral Rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integrands are functions dependent on $x,$ the derivative of this integral is expressible as where the partial derivative $\backslash tfrac$ 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 $b(x\; ...$
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Alternating Series Test
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised 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. For a generalization, see Dirichlet's test. History Leibniz discussed the criterion in his unpublished ''De quadratura arithmetica'' of 1676 and shared his result with Jakob Hermann in June 1705 and with Johann Bernoulli in October, 1713. It was only formally published in 1993. 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 ...
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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, a brand of biscuit *Leibniz Association, a union of German non-university research institutes from various disciplines See also * Leibnitz (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 * Leibniz calculator, a ...

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, a brand of biscuit *Leibniz Association, a uni ... * Leibniz's rule (other) {{mathdab ...
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