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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] |
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] |
Identity Of Indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' is also possessed by ''y'' and vice versa. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below. A form of the principle is attributed to the German philosopher Gottfried Wilhelm Leibniz. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's l ... [...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] |
