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 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 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 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 ) = n ! k ! ( n − k ) ! {displaystyle {n choose k}={n! over k!(n-k)!}} is the binomial coefficient and f ( 0 ) ≡ f . {displaystyle f^{(0)}equiv f.} This can be proved by using the product rule and mathematical induction. If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions: The formula can be generalized to the product of m differentiable functions f1,...,fm. where the sum extends over all m-tuples (k1,...,km) of non-negative integers with ∑ t = 1 m k t = n , {displaystyle sum _{t=1}^{m}k_{t}=n,} and are the multinomial coefficients. This is akin to the multinomial formula from algebra. The proof of the general Leibniz rule proceeds by induction. Let f {displaystyle f} and g {displaystyle g} be n {displaystyle n} -times differentiable functions. The base case when n = 1 {displaystyle n=1} claims that: which is the usual product rule and is known to be true. Next, assume that the statement holds for a fixed n ≥ 1 , {displaystyle ngeq 1,} that is, that