Leibniz rule (generalized product rule)
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See Leibniz rule, a disambiguation page, for other meanings of this term.
In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by
where 
is the usual binomial coefficient.
This can be proved by using the product rule and mathematical induction.
With the multi-index notation the rule states more generally:
which follows from the application of the aforementioned formula to each component of a multi-index.
