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...

In probability theory, the chain rule (also called the general product rule) describes how to calculate the probability of the intersection of, not necessarily...

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...

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}...

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...

\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 )...

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...

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...

{\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...

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

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...

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

( 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 )...

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

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

{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 and g are functions...

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

(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...

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...

functions. For constant rule and sum rule, see Apostol 1967, p. 161, 164, respectively. For the product rule, quotient rule, and chain rule, see Varberg, Purcell...

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...

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...

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

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...

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...

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...

) {\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 )...

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...

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...