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In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule. It states that if f {\displaystyle f} and g {\displaystyle g} are n {\displaystyle n} -times differentiable functions, then the product f g {\displaystyle fg} is also n {\displaystyle n} -times differentiable and its n {\displaystyle n} th derivative is given by
where = n ! k ! ! {\displaystyle {n \choose k}={n! \over k!!}} is the binomial coefficient and f {\displaystyle f^{}} denotes the jth derivative of f = f {\displaystyle f^{}=f} ].
The rule can be proved by using the product rule and mathematical induction.