General Leibniz rule

Short description: Generalization of the product rule in calculus

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 [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are [math]\displaystyle{ n }[/math]-times differentiable functions, then the product [math]\displaystyle{ fg }[/math] is also [math]\displaystyle{ n }[/math]-times differentiable and its [math]\displaystyle{ n }[/math]th derivative is given by

[math]\displaystyle{ (fg)^{(n)}=\sum_{k=0}^n {n \choose k} f^{(n-k)} g^{(k)}, }[/math]

where [math]\displaystyle{ {n \choose k}={n!\over k! (n-k)!} }[/math] is the binomial coefficient and [math]\displaystyle{ f^{(j)} }[/math] denotes the jth derivative of f (and in particular [math]\displaystyle{ f^{(0)}= f }[/math]).

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

Second derivative

If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions:

[math]\displaystyle{ (fg)''(x)=\sum\limits_{k=0}^{2}{\binom{2}{k} f^{(2-k)}(x)g^{(k)}(x)}=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x). }[/math]

More than two factors

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

[math]\displaystyle{ \left(f_1 f_2 \cdots f_m\right)^{(n)}=\sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} \prod_{1\le t\le m}f_{t}^{(k_{t})}\,, }[/math]

where the sum extends over all m-tuples (k₁,...,k_m) of non-negative integers with [math]\displaystyle{ \sum_{t=1}^m k_t=n, }[/math] and

[math]\displaystyle{ {n \choose k_1, k_2, \ldots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!} }[/math]

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 [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] be [math]\displaystyle{ n }[/math]-times differentiable functions. The base case when [math]\displaystyle{ n=1 }[/math] claims that:

[math]\displaystyle{ (fg)'=f'g+fg', }[/math]

which is the usual product rule and is known to be true. Next, assume that the statement holds for a fixed [math]\displaystyle{ n \geq 1, }[/math] that is, that

[math]\displaystyle{ (fg)^{(n)}=\sum_{k=0}^n\binom{n}{k} f^{(n-k)}g^{(k)}. }[/math]

Then,

[math]\displaystyle{ \begin{align} (fg)^{(n+1)} &= \left[ \sum_{k=0}^n \binom{n}{k} f^{(n-k)} g^{(k)} \right]' \\ &= \sum_{k=0}^n \binom{n}{k} f^{(n+1-k)} g^{(k)} + \sum_{k=0}^n \binom{n}{k} f^{(n-k)} g^{(k+1)} \\ &= \sum_{k=0}^n \binom{n}{k} f^{(n+1-k)} g^{(k)} + \sum_{k=1}^{n+1} \binom{n}{k-1} f^{(n+1-k)} g^{(k)} \\ &= \binom{n}{0} f^{(n+1)} g^{(0)} + \sum_{k=1}^{n} \binom{n}{k} f^{(n+1-k)} g^{(k)} + \sum_{k=1}^n \binom{n}{k-1} f^{(n+1-k)} g^{(k)} + \binom{n}{n} f^{(0)} g^{(n+1)} \\ &= \binom{n+1}{0} f^{(n+1)} g^{(0)} + \left( \sum_{k=1}^n \left[\binom{n}{k-1} + \binom{n}{k} \right]f^{(n+1-k)} g^{(k)} \right) + \binom{n+1}{n+1} f^{(0)} g^{(n+1)} \\ &= \binom{n+1}{0} f^{(n+1)} g^{(0)} + \sum_{k=1}^n \binom{n+1}{k} f^{(n+1-k)} g^{(k)} + \binom{n+1}{n+1}f^{(0)} g^{(n+1)} \\ &= \sum_{k=0}^{n+1} \binom{n+1}{k} f^{(n+1-k)} g^{(k)} . \end{align} }[/math]

And so the statement holds for [math]\displaystyle{ n+1, }[/math] 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:

[math]\displaystyle{ \partial^\alpha (fg) = \sum_{ \beta\,:\,\beta \le \alpha } {\alpha \choose \beta} (\partial^{\beta} f) (\partial^{\alpha - \beta} g). }[/math]

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 [math]\displaystyle{ R = P \circ Q. }[/math] Since R is also a differential operator, the symbol of R is given by:

[math]\displaystyle{ R(x, \xi) = e^{-{\langle x, \xi \rangle}} R (e^{\langle x, \xi \rangle}). }[/math]

A direct computation now gives:

[math]\displaystyle{ R(x, \xi) = \sum_\alpha {1 \over \alpha!} \left({\partial \over \partial \xi}\right)^\alpha P(x, \xi) \left({\partial \over \partial x}\right)^\alpha Q(x, \xi). }[/math]

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

• Binomial theorem – Algebraic expansion of powers of a binomial
• Derivation (differential algebra) – Algebraic generalization of the derivative
• Derivative – Instantaneous rate of change (mathematics)
• Differential algebra – Algebraic study of differential equations
• Pascal's triangle – Triangular array of the binomial coefficients in mathematics
• Product rule – Formula for the derivative of a product
• Quotient rule – Formula for the derivative of a ratio of functions