Some Useful Facts About Derivatives

In this section we present some useful identities involving derivatives, which, together with the formulas for the derivatives of simple functions presented in Table 4.1, will enable you to obtain the derivative of almost any function that you will encounter in physical chemistry. 

Many people think that Eq. (4.25) is more convenient to use than Eq. (4.24). 

From this follows the simple but important fact  

This can be deduced by substituting into the definition of the derivative, or by using Eqs. (4.21) and (4.26). 

The function / is sometimes referred to as a composite junction. It is a function of X, because x is a function of u. Specifying a value of x specifies a value of u. 

---

Section 4.4 Some Useful Facts About Derivatives... 

Recent advances on material s chemistiy have contributed to develop highly featured carbon nanostructures with unforeseen properties that offer unexpected opportunities in many fields. " The majority of these applications derives from the flexible coordination chemistiy of carbon atoms and their unique ability to bind other heteroatoms (either on the surface or inserted within the structural framework) or other carbon atoms, resulting in a wide spectrum of materials and allotropic forms, as shown in Fig. 1. The main carbon allotropes of interest in electrochemical applications and frequently used carbon electrodes are diamond, graphite, graphene and carbon nanotubes. A detailed review on carbon materials structure can be found in ref. 15,17 and 18, and below we summarize some important facts about their characteristics ... 

However praiseworthy this may be, it is still difficult to portray the various maturation paths of coal (or, in other instances, petroleum) as occurring regularly so that, for example, the heteroatoms are scattered evenly throughout the model structure. Nature does not always behave in such regular fashion and there may be a tendency to concentrate, say, the heteroatoms only in certain parts of a molecule. Recognition of this fact alone detracts markedly from, and even nullifies, many of the structures that have been proposed for coal. For the present purposes, it is sufficient to note that spectroscopic methods of estimating the structure of coal also (as do chemical methods of structural determination) suffer from several, sometimes severe, limitations. Thus, while some useful information can be derived about the structure of coal, the concept that the data will eventually lead to an average or even to a representative structure can only be cautiously (if at all) accepted. 

Some of the relevant facts about the angular dispersion of a grating may be derived from the following equation, which is a close approximation when the grating is used in the Littrow configuration small in Figure 2-12) ... 

The most basic method for the determination of the methylxanthines is ultraviolet (UV) spectroscopy. In fact, many of the HPLC detectors that will be mentioned use spectroscopic methods of detection. The sample must be totally dissolved and particle-free prior to final analysis. Samples containing more than one component can necessitate the use of extensive clean-up procedures, ajudicious choice of wavelength, the use of derivative spectroscopy, or some other mathematical manipulation to arrive at a final analytical measurement. A recent book by Wilson has a chapter on the analysis of foods using UV spectroscopy and can be used as a suitable reference for those interested in learning more about this topic.1... 

---