Calculus derivative rules

Some problems in functional optimization can be solved analytically. A topic known as the calculus of variations is included in most courses in advanced calculus. It provides ground rules for optimizing integral functionals. The ground rules are necessary conditions analogous to the derivative conditions (i.e., df jdx = 0) used in the optimization of ordinary functions. In principle, they allow an exact solution but the solution may only be implicit or not in a useful form. For problems involving Arrhenius temperature dependence, a numerical solution will be needed sooner or later. 

It was already assumed in Chapter 1 that readers are familiar with the methods for determining the derivatives of algebraic functions. The general rules, as proven in all basic calculus courses, can be summarized as follows. 

In order to propagate the uncertainties on (143Nd/144Nd)s and (147Sm/144Nd)s towards TDM, we first need to compute the partial derivatives of TDM relative to these two variables. Using the rules of calculus, we get... 

The Euler criterion is therefore equivalent to the familiar mixed partials of a function are equal rule of calculus. This cross-differentiation rule is also the condition for the function z(x, y) to have well-defined (single-valued) first derivatives at each point, and thus to be graphable. 

The concept of limit seems to be essential in the understanding and the present teaching of Calculus. In this article, however, we show how to structure and use differential calculus without introducing this concept. The crucial idea in this development is to use Leibniz rule for the derivative of a product of two functions as one of the postulates, rather than as a derived theorem. Within this approach, the idea of limit could be introduced belatedly and only in order to define concepts such as continuity and differentiability in a more rigorous fashion. 

The rule to calculate the derivative of a product of two functions was first introduced by Leibniz [5, 6], The crux of our presentation is to take the multiplication rule as an initial postulate, rather than as a derived result. Leibniz rule for the derivative of a product of functions is not privy of calculus. It also appears when calculating commutators of matrices or linear operators . ..,BC = B[...,C] + [...,B]C. There is no need to invoke the concept of limit in this case, or when dealing with Lie brackets, or other derivations. The ultimate justification for this choice of initial postulate is given a posteriori in terms of the logarithmic function [7]. 

This rule is the core of differential calculus. We will actually prove that it allows us to interpret the derivative of a function at a point (x, y) as the slope of the tangent line touching it. We shall henceforth refer to the latter simply as the slope of the curve. 

These three rules define the differential calculus without involving the concept of limit. The derivative of positive powers and of polynomials follow directly from a straightforward application of these rules. The case of the... 

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

The procedure based on the multiplication rule as an initial postulate can also be generalized to the derivative of complex functions of a complex variable. Vector calculus would benefit from this approach, since the V operator also obeys Leibniz rule. In both of these cases we would have to generalize the basic Rule 1 and make it consistent with the corresponding case. 

If z(x, y), then x(y, z) and y(z, x). Each of these functions permits the definition of two partial derivatives. What are the relationships between the six partial derivatives formed from three variables If two partial derivatives hold the same variable constant, we only have two variables and can use the reciprocal rule from ordinary calculus ... 

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... 

This should be a simple question, but for some reason many students erase all knowledge of the rules of differentiation as soon as they complete their last calculus class Simply derive the following equation for the chemical potential of the solvent in a polymer solution from the Floiy-Huggins equation ... 

A transport equation for the turbulent kinetic energy, or actually the momentum variance, can be derived by multiplying the equation for the fluctuating component v[, (1.389), by 2u, thereafter use the product rule of calculus to convert some of the terms in the provisional equation, and Anally time average the resulting equation [154]. 

Prom this expression two of the unknown derivatives in the ODE (1.417) can be deduced by simple rules of calculus ... 

Vi = d/dri. In deriving Eq. (B.106) we employed the space representation of the Hamiltonian operator [see Eq. (2.95)], and the fact that the classic Hamiltonian function can be split into kinetic- and potential-energy contributions iiccording to Eq. (2.100). Terms proportional to in Eq. (B.106) arise from the kinetic part of // applied to the product of terms on the right side of Eq. (B.104) (using, of course, the product rule of conventional calculus). 

Finally, the partial derivative dVfdx may be evaluated from differential expressions such as (2.6) using the chain rule of elementary calculus. From (2.5), F is a function of T and P, or V = V(T, P). Let T and P each be functions of two other variables X and y,... 

Show the derivation of Eqs. 10,19 and 10,20. Hint These transformations are made in terms of the chain rules of advanced calculus, as shown in any text on advanced calculus. For example, Eq. 10.20 is worked out in detail by Kaplan [9]. Show that the vorticity (Eq. 10.34) in polar coordinates is given by = VJr -I- dVgldr - (1 /r)(dVJdd). Then, using the polar coordinate form, show that the flow described by Eq. 10.30 is not irrotational, but that the flow described by Eq. 10.36 is irrotational. 

We present the chain rule here, but do not derive it. Derivations can be found in most calculus books. 

We focus on a single dimension, the x dimension, and determine the function gx v ). If we were to take the derivative of equation 19.16 with respect to v, we would still have an equality. The problem is that the left side and the right side of equation 19.16 are written in terms of different variables. However, calculus has something called the chain rule, and equation 19.17 gives us a relationship between V and Vj5, the two variables of interest here. Taking the derivative of each side of equation 19.16 with respect to vy. 

We can check the relative tail weights exactly. Since we have formulas for the target and the starting densities, we can take the derivative of their ratio. Using the limit theorems of calculus and L Hospital s rule we can find limits of this ratio for 0 approaching oo. The candidate distribution has heavier tails if the limits of the ratio are greater than 1. 

E is said to be a functional of the functional E = E[W] takes a value that depends on the functional form of W, not merely on particular numerical values of a set of independent variables (adjustable parameters). The analysis of the behaviour of E[W] with respect to variation of W is functional analysis and parallels the study of the behaviour of f(x) with respect to variation of x (i.e. analysis in the usual sense) thus, for example, it is possible to discuss variations in terms of functional derivatives SEISfP. We shall, however, make little use of such concepts, in spite of their formal value (see e.g. Feynman and Hibbs, 1965), since variation functions are most commonly defined in terms of a finite number of parameters, to which the ordinary rules of differential calculus can be applied. The other problems we meet can also be solved by elementary methods. 

There are many uses for differential calculus in physical chemistry however, before going into these, let us first review the mechanics of differentiation. The functional dependence of the variables of a system may appear in many different forms as first- or second-degree equations, as trigonometric functions, as logarithms or exponential functions. For this reason, consider the derivatives of these types of functions that are used extensively in physical chemistry. Also included in the list below are rules for differentiating sums, products, and quotients. In some cases, examples are given in order to illustrate the application to physicochemical equations. 

---