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generalizations of the Leibniz rule (Theorem)

For the derivative, the product rule

$\displaystyle (fg)' = f'g + fg' $
is known as the Leibniz rule. Below are various ways it can be generalized.

Higher derivatives

Let $ f,g$ be real (or complex) functions defined on an open interval of $ \mathbb{R}$. If $ f$ and $ g$ are $ k$ times differentiable, then
$\displaystyle (fg)^{(k)} = \sum_{r=0}^k {k \choose r} f^{(k-r)} g^{(r)}. $

Generalized Leibniz rule for more functions

Let $ f_1,\ldots,f_r$ be real (or complex) valued functions that are defined on an open interval of $ \mathbb{R}$. If $ f_1,\ldots,f_r$ are $ n$ times differentiable, then
$\displaystyle \frac{d^n}{dt^n}\prod_{i=1}^rf_i(t) = \sum_{n_1+\cdots+n_r=n} {n \choose n_1,n_2,\ldots,n_r} \prod_{i=1}^r \frac{d^{n_i}}{dt^{n_i}}f_i(t). $
where $ {n \choose n_1,n_2,\ldots,n_r}$ is the multinomial coefficient.

Leibniz rule for multi-indices

If $ f,g:\mathbb{R}^n \to \mathbb{R}$ are smooth functions defined on an open set of $ \mathbb{R}^n$, and $ j$ is a multi-index, then
$\displaystyle \partial^j(fg) = \sum_{i\le j} {j \choose i} \partial^i(f)\, \partial^{j-i}(g),$
where $ i$ is a multi-index.

Bibliography

1
Leibniz, Gottfried W. Symbolismus memorabilis calculi Algebraici et Infinitesimalis, in comparatione potentiarum et differentiarum; et de Lege Homogeneorum Transcendentali, Miscellanea Berolinensia ad incrementum scientiarum, ex scriptis Societati Regiae scientarum pp. 160-165 (1710). Available online at the digital library of the Berlin-Brandenburg Academy.



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"generalizations of the Leibniz rule" is owned by GeraW. [ full author list (5) ]
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See Also: multinomial theorem, n'th derivative of a determinant

Other names:  Leibniz rule

Attachments:
proof of generalized Leibniz rule (Proof) by rspuzio
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Cross-references: multi-index, open set, smooth functions, multinomial coefficient, differentiable, open interval, functions, complex, real, product rule, derivative
There are 9 references to this entry.

This is version 10 of generalizations of the Leibniz rule, born on 2004-07-28, modified 2007-04-04.
Object id is 6042, canonical name is GeneralizedLeibnizRule.
Accessed 6143 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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