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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,
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.
Differentiation is linear. The product rule. The chain rule. The inverse function rule. Power laws, polynomials, quotients, and reciprocals. The polynomial or elementary power rule. The reciprocal rule. The quotient rule. Generalized power rule.
Existential generalization / instantiation. In classical logic, a hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. Ancient references point to the works of Theophrastus and Eudemus for the first investigation of this kind of syllogisms.
Evaluating at a vector in measures how much points in the th coordinate direction. 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 ...
v. t. e. Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno ( 1855, 1857 ), although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast had ...
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. (The latter equality only holds if A ( t) is invertible .)
t. e. 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 ...
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