Chain rule, functional derivatives

The proof of this derivative is slightly complicated but for our purposes we can memorize it. It is instmctive here to introduce the concept of the CHAIN RULE for derivatives of functions, which are in turn functions of variables as for instance when m = ax in e = e°. Then we have to take the derivative with respect to u and the chain derivative of u with respect to x so we have ... 

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... 

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. 

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... 

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... 

Optimization of the molecular geometry at the HF level appears at first sight to be a daunting task because of the difficulty of obtaining analytic derivatives (see Section 2.4.1). To take the first derivative of Eq. (4.54) with respect to the motion of an atom, we can exhaustively apply the chain rule term by term. Thus, we must determine derivatives of basis functions... 

Quite frequently we are faced with the problem of differentiating functions of functions, such as y — ln(x2 + x +1). The derivative of this function is not immediately obvious, and so we use a strategy known as the chain rule to reduce the problem to a more manageable form. We can proceed as follows ... 

We have dealt with the sum and the product of functions, as well as the quotient. In order to calculate the derivative of the composition of two functions, we take advantage of the fact that the derivative is defined as the quotient of two differentials. In the case in which / is a function of u, in turn a function of x, the chain rule will tell us how to calculate the derivative of f(u(x)) with respect to x simply as ... 

The more usual applications of the chain rule imply breaking up the function whose derivative we want to compute into smaller portions, where each piece is itself a function of the independent variable. As a simple example, let us calculate the derivative of a rational function such as ... 

The chain rule is an indispensable tool in differential calculus. It allows for the simplification of derivatives of composite functions. 

Relation 4 The derivative of complicated functions can be reduced to the derivatives of simpler function by the chain rule ... 

This defines a Gateaux functional derivative [26, 102], whose value depends on a direction in the function space, reducing to a Frechet derivative only if all e, are equal. Defining Tt = t + v, an explicit orbital index is not needed if Eq. (5.10) is interpreted to define a linear operator acting on orbital wave functions, TL — v = t. The elementary chain rule is valid when the functional... 

Since multilayer perceptions use neurons that have differentiable functions, it was possible, using the chain rule of calculus, to derive a delta rule for training similar in form and function to that for perceptions. The result of this clever mathematics is a powerful and relatively efficient iterative method for multilayer perceptions. The rule for changing weights into a neuron unit becomes... 

By virtue of the functional chain rule, the functional derivative of p, with respect to Pext provides a link of the interacting response function (145) to its noninteracting counterpart ... 

Making use of the functional chain rule once more to calculate the functional derivative of with respect to Pe, one gets... 

Most research focuses on the time rate of change of these stocks, stratified by vegetation type. Using the chain rule (see appendix), we can derive the following relationship where the time rate of change of carbon in the landscape unit is a function of four groups of fluxes. These fluxes are listed as a-d for each of the vegetation types in the landscape unit ... 

To respect the orientation of frames, we will make also x = y and X3 = x. By applying the chain rule for multivariate functions to the derivatives of Eqn. (4), we obtain ... 

The negative sign in the definition of V follows the standard convention in physics it implies that the particle always moves downhill as the motion proceeds. To see this, we think of x as a function of t, and then calculate the time-derivative of V(x(r)). Using the chain rule, we obtain... 

The extended form of the Leibnitz s rule can be derived by use of the phase indicator function (3.260), the topological equation (3.269), the Reynolds averaging rules (1.379) and the chain rule [54] ... 

Before turning to functional Taylor expansion, we note that many operations with ordinary derivatives can be extended to functional derivatives. We note, in particular, the chain rule of differentiation. 

Next, we apply the chain rule of functional derivatives (see Appendix B), which, for the present case, takes the form... 

Using the chain rule for the functional derivative (Appendix B), we find... 

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