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General Leibniz rule

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In calculus, the general Leibniz rule, named after Gottfried Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if f and g are n -times differentiable functions, then the product f g is also n -times differentiable and its n th derivative is given by

Contents

( f g ) ( n ) ( x ) = ∑ k = 0 n ( n k ) f ( n − k ) ( x ) g ( k ) ( x )

where ( n k ) = n ! k ! ( n − k ) ! is the binomial coefficient and f ( 0 ) ( x ) = f ( x ) .

This can be proved by using the product rule and mathematical induction (see proof below).

Second derivative

In case n = 2 :

( f g ) ″ ( x ) = ∑ k = 0 2 ( 2 k ) f ( 2 − k ) ( x ) g ( k ) ( x ) = f ″ ( x ) g ( x ) + 2 f ′ ( x ) g ′ ( x ) + f ( x ) g ″ ( x ) .

The binomial coefficients can be deduced thanks to the Pascal's triangle.

More than two factors

The formula can be generalized to the product of m differentiable functions f1,...,fm.

( f 1 f 2 ⋯ f m ) ( n ) = ∑ k 1 + k 2 + ⋯ + k m = n ( n k 1 , k 2 , … , k m ) ∏ 1 ≤ t ≤ m f t ( k t ) ,

where the sum extends over all m-tuples (k1,...,km) of non-negative integers with ∑ t = 1 m k t = n and

( n k 1 , k 2 , … , k m ) = n ! k 1 ! k 2 ! ⋯ k m !

are the multinomial coefficients. This is akin to the multinomial formula from algebra.

Multivariable calculus

With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally:

∂ α ( f g ) = ∑ { β : β ≤ α } ( α β ) ( ∂ β f ) ( ∂ α − β g ) .

This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let P and Q be differential operators (with coefficients that are differentiable sufficiently many times) and R = P ∘ Q . Since R is also a differential operator, the symbol of R is given by:

R ( x , ξ ) = e − ⟨ x , ξ ⟩ R ( e ⟨ x , ξ ⟩ ) .

A direct computation now gives:

R ( x , ξ ) = ∑ α 1 α ! ( ∂ ∂ ξ ) α P ( x , ξ ) ( ∂ ∂ x ) α Q ( x , ξ ) .

This formula is usually known as the Leibniz formula. It is used to define the composition in the space of symbols, thereby inducing the ring structure.

References

General Leibniz rule Wikipedia
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