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In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.
In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there are a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions. [1] [2]
The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form. Cross product rule
The rule can be thought of as an integral version of the product rule of differentiation; it is indeed derived using the product rule. The integration by parts formula states: ′ = [() ()] ′ () = () () ′ ().
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables.
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
General Leibniz rule – Generalization of the product rule in calculus; Integration by parts – Mathematical method in calculus; Inverse functions and differentiation – Calculus identity; Product rule – Formula for the derivative of a product