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...

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...

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 may refer to: Chain rule in calculus: d y d x = d y d u ⋅ d u d x . {\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\mathrm {d}...

portal Chain rule Differentiation of integrals Leibniz rule (generalized product rule) Reynolds transport theorem, a generalization of Leibniz rule Protter...

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...

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...

(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...

It has a similar form to chain rule in probability theory, except that addition instead of multiplication is used. Bayes' rule for conditional entropy...

\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 )...

and is therefore an instance of a vector-valued differential form. The chain rule has a particularly elegant statement in terms of total derivatives. It...

{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...

( 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}}\...

vector field. We have the following special cases of the multi-variable chain rule. ∇ ( f ∘ ϕ ) = ( f ′ ∘ ϕ ) ∇ ϕ ( r ∘ f ) ′ = ( r ′ ∘ f ) f ′ ( ϕ ∘ r )...

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...

{\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...

in computing parameter updates. It is an efficient application of the chain rule to neural networks. Backpropagation computes the gradient of a loss function...

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 Ω ′ ⊆ C...

and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, partial derivatives of arbitrary order...

respect to time or one of the other variables requires application of the chain rule, since most problems involve several variables. Fundamentally, if a function...

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...

rules Derivative of a constant Sum rule in differentiation Constant factor rule in differentiation Linearity of differentiation Power rule Chain rule...

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...

different operations, as can be seen when considering differentiation (chain rule) or integration (integration by substitution). A very simple example of...

stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion...

) {\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 )...

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...

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...

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...

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...