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The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions. The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be
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. [1] [2] [3] For a generalization, see Dirichlet's test. [4] [5] [6]
In calculus, the product rule (or Leibniz rule [1] 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 () ′ = ′ + ′ or in Leibniz's notation as () = +.
These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in: Mathematical Handbook of Formulas and Tables (3rd edition) , S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISBN 978-0-07-154855-7 .
Leibniz' law may refer to: The product rule; General Leibniz rule, a generalization of the product rule; Identity of indiscernibles; See also. Leibniz (disambiguation)
Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: Product rule in differential calculus; General Leibniz rule, ...
The Augusta Rule refers to Internal Revenue Code Section 280(A), which allows owners to rent out their property for 14 days or less in a year without reporting the income they earn. Since the ...
Leibniz theorem (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; The Fundamental theorem of calculus, also called Newton-Leibniz theorem.