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 $f$ and $g$ are $n$-times differentiable functions, then the product $fg$ is also $n$-times differentiable and its $n$th derivative is given by

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

where $=$ is the binomial coefficient and $f^$ denotes the ''j''th derivative of ''f'' (and in particular $f^= f$).

The rule can be proved 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 ''f''₁,...,''f''_''m''.
:$\left(f_1 f_2 \cdots f_m\right)^=\sum_ \prod_f_^\,,$
where the sum extends over all ''m''-tuples (''k''₁,...,''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 $f$ and $g$ be $n$-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 $n \geq 1,$ that is, that

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

Then,

:\begin
(fg)^ &= \left \sum_^n \binom f^ g^ \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 fg^ \\
&= \binom f^ g + \left( \sum_^n \left binom + \binom \right^ g^ \right) + \binom fg^ \\
&= \binom f^ g + \sum_^n \binom f^ g^ + \binomfg^ \\
&= \sum_^ \binom f^ g^ .
\end

And so the statement holds for $n+1,$ 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 $R = P \circ Q.$ 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.

See also

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References

{{Calculus topics

Articles containing proofs
Differentiation rules
Gottfried Wilhelm Leibniz
Mathematical identities
Theorems in analysis
Theorems in calculus