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generalizations of the Leibniz rule
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(Theorem)
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For the derivative, the product rule $$ (fg)' = f'g + fg' $$ is known as the Leibniz rule. Below are various ways it can be generalized.
Let $f,g$ be real (or complex) functions defined on an open interval of $\sR$ If $f$ and $g$ are $k$ times differentiable, then $$ (fg)^{(k)} = \sum_{r=0}^k {k \choose r} f^{(k-r)} g^{(r)}. $$
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 $$ \frac{d^n}{dt^n}\prod_{i=1}^rf_i(t) = \sum_{n_1+\cdots+n_r=n} {n \choose n_1,n_2,\ldots,n_r} \prod_{i=1}^r \frac{d^{n_i}}{dt^{n_i}}f_i(t). $$ where ${n \choose n_1,n_2,\ldots,n_r}$ is the multinomial coefficient.
If $f,g:\sR^n \to \sR$ are smooth functions defined on an open set of $\sR^n$ and $j$ is a multi-index, then $$ \partial^j(fg) = \sum_{i\le j} {j \choose i} \partial^i(f)\, \partial^{j-i}(g),$$ where $i$ is a multi-index.
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- 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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Cross-references: multi-index, open set, smooth functions, multinomial coefficient, differentiable, open interval, functions, complex, real, product rule, derivative
There are 12 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 10130 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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