General Leibniz rule

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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In calculus, the general Leibniz rule,^[1] 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 n-times differentiable functions, then the product ${\displaystyle fg}$ is also n-times differentiable and its 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 ${\displaystyle f^{(0)}=f}$).

The rule can be proven by using the product rule and mathematical induction.