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Formula in calculus
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions z and y in terms of the derivatives
Chain_rule
Technique in integral evaluation
reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation
Integration_by_substitution
Topics referred to by the same term
The chain rule in calculus is the formula for the derivative/differentiation of a composition of functions or operations. Chain rule may also refer to
Chain_rule_(disambiguation)
Probability theory concept
In probability theory, the chain rule (also called the general product rule) describes how to calculate the probability of the intersection of, not necessarily
Chain_rule_(probability)
Differentiation under the integral sign formula
portal Chain rule Differentiation of integrals Leibniz rule (generalized product rule) Reynolds transport theorem, a generalization of Leibniz rule Protter
Leibniz_integral_rule
Formula for the derivative of a matrix determinant
(A_{11},A_{12},\ldots ,A_{21},A_{22},\ldots ,A_{nn})} so that, by the chain rule, its differential is d det ( A ) = ∑ i ∑ j ∂ F ∂ A i j d A i j . {\displaystyle
Jacobi's_formula
Type of derivative in mathematics
the chain rule takes such dependencies into account. Write γ ( x ) = ( x , y ( x ) ) {\displaystyle \gamma (x)=(x,y(x))} . Then, the chain rule says
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Formula for the derivative of a product
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions
Product_rule
Formula for the derivative of a ratio of functions
g(x)-1\cdot g'(x)}{g(x)^{2}}}={\frac {-g'(x)}{g(x)^{2}}}.} Utilizing the chain rule yields the same result. Let h ( x ) = f ( x ) g ( x ) {\displaystyle
Quotient_rule
Rules for computing derivatives of functions
{df}{dx}}.} The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule. If f {\textstyle f} and
Differentiation_rules
Measure of relative information in probability theory
It has a similar form to chain rule in probability theory, except that addition instead of multiplication is used. Bayes' rule for conditional entropy
Conditional_entropy
Mathematical process of finding the derivative of a trigonometric function
\theta \,.} To compute the derivative of the cosine function from the chain rule, first observe the following three facts: cos θ = sin ( π 2 − θ )
Differentiation of trigonometric functions
Differentiation_of_trigonometric_functions
Formula for the derivative of an inverse function
( y ) = x {\displaystyle f^{-1}(y)=x} in terms of x and applying the chain rule, yielding that: d x d y d y d x = d x d x {\displaystyle {\frac {dx}{dy}}\
Inverse_function_rule
Concept in complex analysis
with respect to z ¯ {\displaystyle {\bar {z}}} using the multivariable chain rule, treating x , y {\displaystyle x,y} as intermediate variables. Definition
Wirtinger_derivatives
Optimization algorithm for artificial neural networks
in computing parameter updates. It is an efficient application of the chain rule to neural networks. Backpropagation efficiently computes the gradient
Backpropagation
Method of differentiating single-term polynomials
{\displaystyle f'(x)=f(x)=e^{x}} , as was required. Therefore, applying the chain rule to f ( x ) = e r ln x {\displaystyle f(x)=e^{r\ln x}} , we see that
Power_rule
Numerical calculations carrying along derivatives
and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, partial derivatives of arbitrary order
Automatic_differentiation
Mathematical identities
vector field. We have the following special cases of the multi-variable chain rule. ∇ ( f ∘ ϕ ) = ( f ′ ∘ ϕ ) ∇ ϕ {\displaystyle \nabla (f\circ \phi )=\left(f'\circ
Vector_calculus_identities
Notion in statistics
and therefore ∫ f d x = 1 {\displaystyle \int f\,dx=1} . By using the chain rule on the partial derivative of log f {\displaystyle \log f} and then dividing
Fisher_information
Instantaneous rate of change (mathematics)
functions. For constant rule and sum rule, see Apostol 1967, pp. 161, 164, respectively. For the product rule, quotient rule, and chain rule, see Varberg, Purcell
Derivative
Mathematical technique for simplification
different operations, as can be seen when considering differentiation (chain rule) or integration (integration by substitution). A very simple example of
Change_of_variables
Branch of mathematics
of the fundamental theorem of calculus around 1670. The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic
Calculus
Manifold upon which it is possible to perform calculus
being able to compute the partial derivatives of the LHS applying the chain rule to the RHS. The same problem is found if one considers instead functions
Differentiable_manifold
Identity in Itô calculus analogous to the chain rule
stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion
Itô's_lemma
Lower bound for size of software program
The chain rule[citation needed] for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states: H ( X , Y ) = H ( X )
Chain rule for Kolmogorov complexity
Chain_rule_for_Kolmogorov_complexity
Information theory
) {\displaystyle I(X;Y|Z)=I(X;Y,Z)-I(X;Z)} usually rearranged as the chain rule for mutual information I ( X ; Y , Z ) = I ( X ; Z ) + I ( X ; Y | Z )
Conditional mutual information
Conditional_mutual_information
Relation between relative derivatives of three variables
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule, Euler's chain rule, or the reciprocity theorem, is a
Triple_product_rule
Problems that make use of the relations to rates of change
respect to time or one of the other variables requires application of the chain rule, since most problems involve several variables with respect to the input
Related_rates
Matrix of partial derivatives of a vector-valued function
Composable differentiable functions f : Rn → Rm and g : Rm → Rk satisfy the chain rule, namely J g ∘ f ( x ) = J g ( f ( x ) ) J f ( x ) {\displaystyle \mathbf
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Assignment of a tensor continuously varying across a region of space
an advanced explanation of the tensor concept, one can interpret the chain rule in the multivariable case, as applied to coordinate changes, also as the
Tensor_field
Mathematical rule for inverting probabilities
theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (/beɪz/), gives a mathematical rule for inverting conditional probabilities,
Bayes'_theorem
Calculus on stochastic processes
main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma. This enables problems to be
Stochastic_calculus
Calculus of stochastic differential equations
Stratonovich integral as an alternative formulation; it does follow the chain rule, and does not require Itô's lemma. The two integral forms can be converted
Itô_calculus
Real function with finite total variation
calculus for BV functions: in the paper (Vol'pert 1967) he proved the chain rule for BV functions and in the book (Hudjaev & Vol'pert 1985) he, jointly
Bounded_variation
Multivariate derivative (mathematics)
a ) . {\displaystyle \nabla (fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).} Chain rule Suppose that f : A → R is a real-valued function defined on a subset A
Gradient
Mathematical method in calculus
using a combination of the inverse chain rule method and the natural logarithm integral condition. The LIATE rule is a rule of thumb for integration by parts
Integration_by_parts
Distance along a curve
\mathbf {C} \right)'(t)\right|.} Evaluating the derivative requires the chain rule for vector fields: D ( x ∘ C ) = ( x u x v ) ( u ′ v ′ ) = x u u ′ +
Arc_length
and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be
Glossary_of_calculus
Gradient descent learning rule in machine learning
w_{ji}}}} To find the left derivative, we simply apply the power rule and the chain rule: ∂ E ∂ w j i = − ( t j − y j ) ∂ y j ∂ w j i {\displaystyle {\frac
Delta_rule
Concept in information processing
{\displaystyle X\rightarrow Z\rightarrow Y} also forms a Markov chain. One can apply the chain rule for mutual information to obtain two different decompositions
Data_processing_inequality
rules Derivative of a constant Sum rule in differentiation Constant factor rule in differentiation Linearity of differentiation Power rule Chain rule
List_of_calculus_topics
Theorem in mathematics
{\displaystyle f^{-1}} were differentiable at b {\displaystyle b} , then, by the chain rule, 1 = ( f − 1 ∘ f ) ′ ( a ) = ( f − 1 ) ′ ( b ) f ′ ( a ) {\displaystyle
Inverse_function_theorem
Quantum-mechanical framework for simulating molecules and solids
implicit dependence through the KS orbitals. That motivates the use of the chain rule v x c ( r ) = ∫ d r ′ ∑ s [ δ E x c [ { ϕ s } ] δ ϕ s ( r ′ ) δ ϕ s (
Optimized effective potential method
Optimized_effective_potential_method
Mathematical notion of infinitesimal difference
formulae for Stieltjes integral correspond, respectively, to the chain rule and product rule for the differential. Infinitesimal quantities played a significant
Differential_(mathematics)
Mathematical operation in calculus
ln f(x), or the natural logarithm of f. This follows directly from the chain rule: d d x ln f ( x ) = 1 f ( x ) d f ( x ) d x {\displaystyle {\frac {d}{dx}}\ln
Logarithmic_derivative
Method of mathematical differentiation
of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm
Logarithmic_differentiation
Method for deriving motion equations using calculus
respect to crank angle (take second derivative, using the chain rule and the quotient rule): x ″ = d 2 x d A 2 = − r ⋅ cos A − r 2 ⋅ cos 2 A l 2 −
Piston_motion_equations
Class of nonparametric methods
[\varphi (Y)\otimes \varphi (Y)].} In practical implementations, the kernel chain rule takes the following form C ^ X Y π = C ^ X ∣ Y C ^ Y Y π = Υ ( G + λ I
Kernel embedding of distributions
Kernel_embedding_of_distributions
Chronological legal documentation process
accessed 3 November 2020 Federal Rules of Evidence Rule 901 (Authentication and Identification Rule used for Chain of Custody) Non-Federal Chain of Custody
Chain_of_custody
Notion in calculus
They imply the power rule d ( f n ) = n f n − 1 d f {\displaystyle d(f^{n})=nf^{n-1}df} In addition, various forms of the chain rule hold, in increasing
Differential_of_a_function
Measure for evaluating probabilistic forecasts
is strictly proper for distributions with finite first moment, if the chain rule is included in the conditional specification, meaning that there exists
Scoring_rule
Integral used in physics
Unlike the Itô calculus, Stratonovich integrals are defined such that the chain rule of ordinary calculus holds. Perhaps the most common situation in which
Stratonovich_integral
Study of rates of change
general rules rather than directly from the limit definition. These include the sum rule, product rule, quotient rule, and chain rule. The power rule, for
Differential_calculus
Derivative defined on normed spaces
= D f ( x ) + D g ( x ) . {\displaystyle D(f+g)(x)=Df(x)+Dg(x).} The chain rule is also valid in this context: if f : U → Y {\displaystyle f:U\to Y} is
Fréchet_derivative
Rate of change of velocity
acceleration of a particle moving on a curved path can be written using the chain rule of differentiation for the product of two functions of time as: a = d
Acceleration
Test for infinite series of monotonous terms for convergence
To see the convergence of the series (5), note that by the power rule, the chain rule, and the above result, − d d x 1 ε ( ln k ( x ) ) ε = 1 ( ln k
Integral_test_for_convergence
Derivative of a function with multiple variables
differential Symmetry of second derivatives Triple product rule, also known as the cyclic chain rule. Cajori, Florian (1952), A History of Mathematical Notations
Partial_derivative
Scientific area at the interface between computer science and mathematics
operations, like simplification of expressions, differentiation using the chain rule, polynomial factorization, indefinite integration, etc. Computer algebra
Computer_algebra
American industrial engineer, control theorist and mathematician
backpropagation (due to Henry J. Kelley and Arthur E. Bryson) using only the chain rule. He also coauthored Mind Over Machine with his brother Hubert Dreyfus
Stuart_Dreyfus
Generalized chain rule in calculus
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
Faà_di_Bruno's_formula
Theorem in vector calculus
( J ψ ( u , v ) F ) ∂ ψ ∂ u − ∂ ψ ∂ u ⋅ ( J ψ ( u , v ) F ) ∂ ψ ∂ v (chain rule) = ∂ ψ ∂ v ⋅ ( J ψ ( u , v ) F − ( J ψ ( u , v ) F ) T ) ∂ ψ ∂ u {\displaystyle
Stokes'_theorem
Mathematical concept
many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions
Differentiation in Fréchet spaces
Differentiation_in_Fréchet_spaces
Operation in differential geometry
J_{0}^{k}f\circ J_{0}^{k}g=J_{0}^{k}(f\circ g).} It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the
Jet_(mathematics)
Approximation of a function by a polynomial
{1}{x^{2}}}}&x>0\\0&x\leq 0.\end{cases}}\end{aligned}}} Using the chain rule repeatedly by mathematical induction, one shows that for any order k,
Taylor's_theorem
Probabilistic graphical representation of causal relationships
(for true) and F (for false). The joint probability function is, by the chain rule of probability, Pr ( G , S , R ) = Pr ( G ∣ S , R ) Pr ( S ∣ R ) Pr (
Bayesian_network
Ideographic art tradition in parts of Africa
weaving. Various studies suggest that the drawing experts knew specific rules of "chaining" and "elimination" relating to the systematic construction of monolinear
Lusona
Complex-differentiable (mathematical) function
real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule. A function is holomorphic on an open set U {\displaystyle
Holomorphic_function
Details of the rules for the abstract strategy board game for two players
a chain. The basic rules are formulated here in a more detailed way to ease their presentation in § Explanation of the basic rules below. (Each rule and
Rules_of_Go
Probability of an event occurring, given that another event has already occurred
portal Bayes' theorem Bayesian epistemology Borel–Kolmogorov paradox Chain rule (probability) Class membership probabilities Conditional independence
Conditional_probability
Criterion for integration in terms of elementary functions
{\displaystyle F,} in which case, this condition is analogous to the ordinary chain rule. However, F {\displaystyle F} is not necessarily equipped with a unique
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
Syllogism with conditional premise(s)
syllogism is the name of a valid rule of inference (often abbreviated HS and sometimes also called the chain argument, chain rule, or the principle of transitivity
Hypothetical_syllogism
Technique in graph theory
this to sets of σ {\displaystyle \sigma } -flags gives the Chain Rule: Theorem (Chain Rule): If F 1 , F 2 , … , F t , G {\displaystyle F_{1},F_{2},\ldots
Flag_algebra
Structure defining distance on a manifold
{\displaystyle \left\|\cdot \right\|} represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives: r → u
Metric_tensor
Random process independent of past history
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability
Markov_chain
State-dependent measures that converge to the mutual information
P_{X}]} . As a special case of the chain-rule for Kullback-Liebler divergerences, specific-surprise follows the chain-rule for variables. Using Z {\displaystyle
State-dependent_information
Index of articles associated with the same name
There are many terms in mathematics that begin with cyclic: Cyclic chain rule, for derivatives, used in thermodynamics Cyclic code, linear codes closed
Cyclic_(mathematics)
Representation of water movement in unsaturated soils
\theta (h)} , which is known as the water retention curve. Applying the chain rule, the Richards equation may be reformulated as either h {\displaystyle
Richards_equation
Point to which functions converge in analysis
( x ) ) = c . {\displaystyle \lim _{x\to a}f(g(x))=c.} However, this "chain rule" does hold if one of the following additional conditions holds: f(b) =
Limit_of_a_function
Notation of differential calculus
especially helpful when considering partial derivatives. It also makes the chain rule easy to remember and recognize: d y d x = d y d u ⋅ d u d x . {\displaystyle
Notation_for_differentiation
Specialized notation for multivariable calculus
mind the most important rules: the chain rule, product rule and sum rule. The sum rule applies universally, and the product rule applies in most of the
Matrix_calculus
Generalization of the concept of directional derivative
properties, also consequences of the fundamental theorem, include: (The chain rule) d ( G ∘ F ) ( u ; x ) = d G ( F ( u ) ; d F ( u ; x ) ) {\displaystyle
Gateaux_derivative
Model describing the departures from ideality in solutions of electrolytes and plasmas
{\displaystyle R(r)} in favor of r {\displaystyle r} while carrying out the chain rule and substituting R ′ ( r ) = a {\displaystyle {R^{\prime }}(r)=a} , then
Debye–Hückel_theory
Finnish mathematician and computer scientist
function that can be represented as a graph, by recursively applying the chain rule to the building blocks of the function. Linnainmaa published it first
Seppo_Linnainmaa
Equation for fixed point of functional composition
meaning h(a) = a, then either Ψ(a) = 0 (or ∞) or s = 1. Application of the chain rule shows that the eigenvalue s is given by s = h′(a), provided that Ψ(a)
Schröder's_equation
Topics referred to by the same term
up chain in Wiktionary, the free dictionary. A chain is a series of connected links which are typically made of metal. Chain may also refer to: Chain mail
Chain_(disambiguation)
Means of expressing certain extremely large numbers
together form a chain of length n + 1 {\displaystyle n+1} . Any chain represents an integer, according to the six rules below. Two chains are said to be
Conway_chained_arrow_notation
Calculus of functions generalization
h → 0 {\displaystyle h\to 0} . As in the one-variable case, there is Chain rule— Let f {\displaystyle f} be as above and g : Y → Z {\displaystyle g:Y\to
Calculus_on_Euclidean_space
Q-analog of the ordinary derivative
\quad g(x)g(qx)\neq 0.} There is also a rule similar to the chain rule for ordinary derivatives. Let g ( x ) = c x k {\displaystyle
Q-derivative
Definite integral of a scalar or vector field along a path
G , {\displaystyle \mathbf {F} =\nabla G,} then by the multivariable chain rule the derivative of the composition of G and r(t) is d G ( r ( t ) ) d t
Line_integral
Mathematical concept that extends the intuitive idea of gluing in topology
acting on various fibers. Another major point is the relation with the chain rule: the discussion of the way there of constructing tensor fields can be
Descent_(mathematics)
Mathematical function, inverse of an exponential function
evaluates to ln(b) bx by the properties of the exponential function, the chain rule implies that the derivative of logb x is given by d d x log b x = 1
Logarithm
Devices fitted to the tires of vehicles to improve traction on snow and ice
Snow chains, or tire chains, are devices fitted to the tires of vehicles to provide increased traction when driving through snow and ice. Snow chains attach
Snow_chains
Concept in mathematics
{dx}{dt}}} and expressing the second derivative of x {\displaystyle x} via the chain rule as d 2 x d t 2 = d v d t = d x d t d v d x = v d v d x {\displaystyle
Autonomous system (mathematics)
Autonomous_system_(mathematics)
Type of ordinary differential equation
derivative of u x 2 {\displaystyle ux^{2}} by reversing the product rule. Applying the chain rule and integrating both sides with respect to x {\displaystyle
Bernoulli differential equation
Bernoulli_differential_equation
Quader, Saad; Russel, Alexander (2019). The combinatorics of the longest-chain rule: Linear consistency for proof-of-stake blockchains (PDF) (Technical report)
List_of_cryptocurrencies
American cryptographer and co-inventor of the Blockchain
and implemented an on-chain miner voting system. Under the longest-chain rule of Nakamoto Consensus, the community-governed chain—backed by 74% of the
Stuart_Haber
Generalization of straight line to a curved space time
motion from the form which uses proper time as a parameter using the chain rule. Notice that both sides of this last equation vanish when the mu index
Geodesics in general relativity
Geodesics_in_general_relativity
Concept in the analysis of dynamical systems
to assume the equilibrium point occurs at 0 {\displaystyle 0} . By the chain rule, for any function, H : R n → R , {\displaystyle H:\mathbb {R} ^{n}\to
Lyapunov_function
Inference engine in an expert system
Forward chaining is a popular implementation strategy for expert systems, business and production rule systems. The opposite of forward chaining is backward
Forward_chaining
Derivative method in calculus
{1}{\sin x}}\cdot {\frac {\cos x}{\sin x}}=-\csc x\cot x.\end{aligned}}} Chain rule – Formula in calculus Difference quotient – Expression in calculus Differentiation
Reciprocal_rule
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