The chain rule

The chain rule computes joint probabihties from conditional probabilities  

To see why tiiis holds, consider what happens when we expand out die conditional probabihties with tiieir definitions from Bayes rule  

Each of the terms in the numerator cancels out with the previous term in the denominator, leaving us with the simple expression that P x, y, z) equals itself 

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In conjunction with the use of isoparametric elements it is necessary to express the derivatives of nodal functions in terms of local coordinates. This is a straightforward procedure for elements with C continuity and can be described as follows Using the chain rule for differentiation of functions of multiple variables, the derivative of a function in terms of local variables ij) can be expressed as... 

In many applications, derivative operators need to be expressed in spherical coordinates. In converting from cartesian to spherical coordinate derivatives, the chain rule is employed as follows ... 

Very often in experimental sciences and engineering functions and their derivatives are available only through their numerical values. In particular, through measurements we may know the values of a function and its derivative only at certain points. In such cases the preceding operational rules for derivatives, including the chain rule, can be apphed numerically. 

Partial derivatives of and may be related to each other by differentiating (5.187) using the chain rule... 

The chain rule of differentiation applied to (A.8) provides, in rectangular Cartesian coordinates,... 

For the input-to-hidden connections, we use the chain rule to find that... 

An alternative, simpler expression can be obtained as follows. Using the chain rule, the temperature derivative of the volume for constant value of the relaxation time is expressed as... 

Similar expressions can be derived for second spatial derivatives. The final form of the equations that result after a generalized coordinate transformation depends on the degree of differentiation by using the chain rule, i.e. on the treatment of the metrics x, x, and y. For more details we refer to the... 

The expressions for the various vector operators in spherical coordinates can be derived with the use of the chain rule. Thus, for example,... 

This expression is a simple generalization of the argument deve Section 2.7. It, and its extension to functions of any number of is referred to as the chain rule . In many applications it is customs) one or more subscripts to the partial derivatives to specify the one variables that were held constant As an example, Eq. (45) becomes... 

Suppose that 4> x,y,z) is a scalar point function, that is, a scalar function that is uniquely defined in a given region. Under a change of coordinate system to, say, x y z, it will take on another form, although its value at any point remains the same. Applying the chain rule (Section 2.12),... 

Before the results obtained in the previous section can be applied, it is necessary to describe briefly the method of underdetertmned multipliers. Given a function /(xi, 2,..., 0 of variables Xj, for which it is desired to find stationary values, the chain rule leads to the expression... 

Expressing Eqs. (32) and (33) with the use of the chain rule leads to three relations which must be simultaneously satisfied, viz. 

In order to simplify this expression and get rid of the S we are going to do an integration by parts. In the previous expression the derivative is with respect to . To obtain derivatives with respect to x we use the chain rule of differentiation ... 

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

If we apply the chain rule to the definition of the density function /y(y)... 

However, we require the inverse of X(2,3) in the chain rule, so we expand X(2,3) in a Gaussian basis set using inner projection theory, [45]... 

This is then easily differentiated using the chain rule ... 

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