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General Leibniz rule : ウィキペディア英語版 | General Leibniz rule
In calculus, the general Leibniz rule,〔Olver, Applications of Lie groups to differential equations, page 318〕 named after Gottfried Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if ''u'' and ''v'' are ''n''-times differentiable functions, then product ''uv'' is also ''n''-times differentiable and its ''n''th derivative is given by : where is the binomial coefficient. This can be proved by using the product rule and mathematical induction. ==More than two factors== The formula can be generalized to the product of ''m'' differentiable functions ''f''1,...,''f''''m''. : where the sum extends over all ''m''-tuples (''k''1,...,''k''''m'') of non-negative integers with and : are the multinomial coefficients. This is akin to the multinomial formula from algebra.
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