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In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
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.
2.5 Chain rule. 2.6 Dot product rule. 2.7 Cross product rule. 3 Second derivative identities. ... We have the following special cases of the multi-variable chain rule.
This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains ′ and ″ and two maps: ′ and : ″ having natural smoothness requirements.
[1] [2] This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c. Polynomials and functions of the form x a [ edit ]
As in the discrete case the joint differential entropy of a set of random variables is smaller or equal than the sum of the entropies of the individual random variables: (,, …,) = [3]: 253 The following chain rule holds for two random variables:
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. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form f(x, y, z) = 0, so each ...
This is the stochastic calculus version of the change of variables formula and chain rule. It differs from the standard result due to the additional term involving the second derivative of f, which comes from the property that Brownian motion has non-zero quadratic variation.
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