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t. e. 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, The chain rule may also be expressed in ...
Chain rule (probability) 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. This rule allows you to express a joint ...
In mathematics, Itô's lemma or Itô's formula (also called the Itô–Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule.
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes ...
The total derivative is a linear combination of linear functionals and hence is itself a linear functional. The evaluation measures how much points in the direction determined by at , and this direction is the gradient. This point of view makes the total derivative an instance of the exterior derivative. Suppose now that is a vector-valued ...
Calculus. In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, [1] The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself.
In mathematical statistics, the Fisher information (sometimes simply called information[1]) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information.
Jacobi's formula. In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then. where tr (X) is the trace of the matrix X and is its adjugate matrix.
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