قاعدة لايبنتس العامة

In calculus, the general Leibniz rule,^[1] named after Gottfried Wilhelm Leibniz, generalizes the product rule for the derivative of the product of two functions (which is also known as "Leibniz's rule"). It states that if $f$ and $g$ are $n$ -times differentiable functions, then the product $f g$ is also $n$ -times differentiable and its $n$ -th derivative is given by $(f g)^{(n)} = \sum_{k = 0}^{n} (\binom{n}{k}) f^{(n - k)} g^{(k)},$ where $(\binom{n}{k}) = \frac{n!}{k! (n - k)!}$ is the binomial coefficient and $f^{(j)}$ denotes the j-th 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, $n = 2$ , the rule gives an expression for the second derivative of a product of two functions: $(f g)^{″} (x) = \sum_{k = 0}^{2} (\binom{2}{k}) f^{(2 - k)} (x) g^{(k)} (x) = f^{″} (x) g (x) + 2 f^{'} (x) g^{'} (x) + f (x) g^{″} (x) .$

More than two factors

The formula can be generalized to the product of m differentiable functions f₁,...,f_m. ${(f_{1} f_{2} \dots f_{m})}^{(n)} = \sum_{k_{1} + k_{2} + \dots + k_{m} = n} (\binom{n}{k_{1}, k_{2}, \dots, k_{m}}) \prod_{1 \leq t \leq m} f_{t}^{(k_{t})},$ where the sum extends over all m-tuples (k₁,...,k_m) of non-negative integers with $\sum_{t = 1}^{m} k_{t} = n,$ and $(\binom{n}{k_{1}, k_{2}, \dots, k_{m}}) = \frac{n!}{k_{1}! k_{2}! \dots k_{m}!}$ are the multinomial coefficients. This is akin to the multinomial formula from algebra.

Proof

The proof of the general Leibniz rule^[2]^:{{{1}}} proceeds by induction. Let $f$ and $g$ be $n$ -times differentiable functions. The base case when $n = 1$ claims that: $(f g)^{'} = f^{'} g + f g^{'},$ 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 $(f g)^{(n)} = \sum_{k = 0}^{n} (\binom{n}{k}) f^{(n - k)} g^{(k)} .$

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

Relationship to the binomial theorem

The Leibniz rule bears a strong resemblance to the binomial theorem, and in fact the binomial theorem can be proven directly from the Leibniz rule by taking $f (x) = e^{a x}$ and $g (x) = e^{b x},$ which gives

(a + b)^{n} e^{(a + b) x} = e^{(a + b) x} \sum_{k = 0}^{n} (\binom{n}{k}) a^{n - k} b^{k},

and then dividing both sides by $e^{(a + b) x} .$ ^[2]^:69

Multivariable calculus

With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally: $\partial^{α} (f g) = \sum_{β : β \leq α} (\binom{α}{β}) (\partial^{β} f) (\partial^{α - β} g) .$

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: $R (x, ξ) = e^{- ⟨ x, ξ ⟩} R (e^{⟨ x, ξ ⟩}) .$

A direct computation now gives: $R (x, ξ) = \sum_{α} \frac{1}{α!} {(\frac{\partial}{\partial ξ})}^{α} P (x, ξ) {(\frac{\partial}{\partial x})}^{α} Q (x, ξ) .$

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.

References

^ Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006.
^ ^أ ^ب Spivey, Michael Zachary (2019). The Art of Proving Binomial Identities. Boca Raton: CRC Press, Taylor & Francis Group. ISBN 9781351215817.

[1] Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006.

[Spivey-2] أ ^ب Spivey, Michael Zachary (2019). The Art of Proving Binomial Identities. Boca Raton: CRC Press, Taylor & Francis Group. ISBN 9781351215817.

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