General Leibniz rule explained

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

and

are -times differentiable functions, then the product

is also -times differentiable and its -th derivative is given by

(fg)^=\sum_^n f^ g^,

where

{n\choosek}={n!\overk!(n-k)!}

is the binomial coefficient and

f^(j)

denotes the jth derivative of f (and in particular

f⁽⁰⁾=f

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

Second derivative

If, for example,, the rule gives an expression for the second derivative of a product of two functions: $(fg)$ (x)=\sum\limits_^=f(x)g(x)+2f'(x)g'(x)+f(x)g(x).

More than two factors

The formula can be generalized to the product of m differentiable functions f₁,...,f_m. $\left(f_1 f_2 \cdots f_m\right)^=\sum_ \prod_f_^\,,$ where the sum extends over all m-tuples (k₁,...,k_m) of non-negative integers with $\sum_^m k_t=n,$ and $= \frac$ are the multinomial coefficients. This is akin to the multinomial formula from algebra.

Proof

The proof of the general Leibniz rule proceeds by induction. Let

and

-times differentiable functions. The base case when

n=1

claims that:

(fg)' = f'g + fg',

which is the usual product rule and is known to be true. Next, assume that the statement holds for a fixed

n\geq1,

that is, that

(fg)^=\sum_^n\binom f^g^.

Then, $\begin (fg)^ &= \left[\sum_{k=0}^n \binom{n}{k} f^{(n-k)} g^{(k)} \right]' \\ &= \sum_^n \binom f^ g^ + \sum_^n \binom f^ g^ \\ &= \sum_^n \binom f^ g^ + \sum_^ \binom f^ g^ \\ &= \binom f^ g^ + \sum_^ \binom f^ g^ + \sum_^n \binom f^ g^ + \binom f^ g^ \\ &= \binom f^ g^ + \left(\sum_^n \left[\binom{n}{k-1} + \binom{n}{k} \right]f^ g^ \right) + \binom f^ g^ \\ &= \binom f^ g^ + \sum_^n \binom f^ g^ + \binomf^ g^ \\ &= \sum_^ \binom f^ g^ . \end$ And so the statement holds for and the proof is complete.

Multivariable calculus

With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally: $\partial^\alpha (fg) = \sum_ (\partial^ f) (\partial^ g).$

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\circQ.

Since R is also a differential operator, the symbol of R is given by:

R(x, \xi) = e^ R (e^).

A direct computation now gives: $R(x, \xi) = \sum_\alpha \left(\right)^\alpha P(x, \xi) \left(\right)^\alpha Q(x, \xi).$

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.

Notes and References

Book: Olver, Peter J. . 2000 . Applications of Lie Groups to Differential Equations . Springer . 318–319 . 9780387950006 .