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mathematical analysis > calculus > differential calculus > general Leibniz rule

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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 ${\\displaystyle f}$ and ${\\displaystyle g}$ are ${\\displaystyle n}$ -times differentiable functions, then the product ${\\displaystyle fg}$ is also ${\\displaystyle n}$ -times differentiable and its ${\\displaystyle n}$ th derivative is given by

${\\displaystyle (fg)^{(n)}=\\sum _{k=0}^{n}{n \\choose k}f^{(n-k)}g^{(k)},}$

where ${\\displaystyle {n \\choose k}={n! \\over k!(n-k)!}}$ is the binomial coefficient and ${\\displaystyle f^{(j)}}$ denotes the jth derivative of f (and in particular ${\\displaystyle f^{(0)}=f}$ ).

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/General_Leibniz_rule)

differential calculus

règle générale de Leibniz

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