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All extensions of calculus have a chain rule. In most of these, the formula remains the same, though the meaning of that formula may be vastly different. One generalization is to manifolds. In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This ...
The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions f {\displaystyle f} and g {\displaystyle g} , the total derivative of the composite function f ∘ g {\displaystyle f\circ g} at a {\displaystyle a} satisfies
Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables = + = + Substitute dy into dx = [() + ()] + By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero () + = Subtracting the second term and multiplying by its inverse gives the triple ...
Chain rule – For derivatives of composed functions; Differentiation of trigonometric functions – Mathematical process of finding the derivative of a trigonometric function; Differentiation rules – Rules for computing derivatives of functions; General Leibniz rule – Generalization of the product rule in calculus
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
(The chain rule) d ( G ∘ F ) ( u ; x ) = d G ( F ( u ) ; d F ( u ; x ) ) {\displaystyle d(G\circ F)(u;x)=dG(F(u);dF(u;x))} for all u ∈ U {\displaystyle u\in U} and x ∈ X . {\displaystyle x\in X.} (Importantly, as with simple partial derivatives , the Gateaux derivative does not satisfy the chain rule if the derivative is permitted to be ...
Simplest rules Sum rule in integration; Constant factor rule in integration; Linearity of integration; Arbitrary constant of integration; Cavalieri's quadrature formula; Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the ...
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 .