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 and are n-times differentiable functions, then the product is also n-times differentiable and its n-th derivative is given by
The rule can be proven by using the product rule and mathematical induction.
Second derivative edit
If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions:
More than two factors edit
The formula can be generalized to the product of m differentiable functions f1,...,fm.
Proof edit
The proof of the general Leibniz rule proceeds by induction. Let and be -times differentiable functions. The base case when claims that:
Then,
Multivariable calculus edit
With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally:
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 Since R is also a differential operator, the symbol of R is given by:
A direct computation now gives:
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 edit
- 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
References edit
- ^ Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006.