General Leibniz rule

Short description: Generalization of the product rule in calculus

In calculus, the general Leibniz rule,^[1] 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 $f g$ 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 jth 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 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{aligned} (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{aligned}

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^{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. https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/General Leibniz rule. Read more

[1] Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006. https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318.

[1]

v t e Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Fundamental theorem Graph of a function Linear function Mean value theorem Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Orders of approximation (ε, δ)-definition of limit
Differential calculus	Chain rule Derivative Differential Differential equation Differential operator Implicit differentiation Inverse functions and differentiation L'Hôpital's rule Leibniz's rule Logarithmic derivative Mean value theorem Newton's method Notation Leibniz's notation Newton's notation Product rule Quotient rule Regiomontanus' angle maximization problem Related rates Simplest rules Constant factor rule in differentiation Linearity of differentiation Power rule Sum rule in differentiation Stationary point First derivative test Second derivative test Extreme value theorem Maxima and minima Taylor's theorem
Integral calculus	Antiderivative Arc length Constant of integration Differentiation under the integral sign Fundamental theorem of calculus Integral of secant cubed Integral of the secant function Integration by parts Integration by substitution Tangent half-angle substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Simplest rules Constant factor rule in integration Linearity of integration Sum rule in integration Trapezoidal rule Trigonometric substitution
Vector calculus	Curl Directional derivative Divergence Divergence theorem Gradient Gradient theorem Green's theorem Laplacian Stokes' theorem
Multivariable calculus	Curvature Disc integration Divergence theorem Exterior Gabriel's Horn Geometric Hessian matrix Jacobian matrix and determinant Line integral Matrix Multiple integral Partial derivative Shell integration Surface integral Tensor Volume integral
Series	Abel's test Alternating Alternating series test Arithmetico-geometric sequence Binomial Cauchy condensation test Direct comparison test Dirichlet's test Euler–Maclaurin formula Fourier Geometric Harmonic Infinite Integral test for convergence Limit comparison test Maclaurin Power Ratio test Root test Taylor Term test
Special functions and numbers	Bernoulli numbers e (mathematical constant) Exponential function Natural logarithm Stirling's approximation
History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	Differentiation rules List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse trigonometric functions List of integrals of irrational functions List of integrals of logarithmic functions List of integrals of rational functions List of integrals of trigonometric functions List of limits List of mathematical symbols Lists of integrals

Anonymous

Search

General Leibniz rule

Namespaces

More

Page actions

Contents

Second derivative

More than two factors

Proof

Multivariable calculus

See also

References

Navigation

Navigation

Help

googletranslator

Navigation

Wiki tools

Wiki tools

Anonymous

Search

General Leibniz rule

Second derivative

More than two factors

Proof

Multivariable calculus

See also

References

Navigation

Wiki tools

Page tools

Other projects

Categories