First-derivative methods

The Newton-Raphson methods of energy minimization (Berkert and Allinger, 1982) utilize the curvature of the strain energy surface to locate minima. The computations are considerably more complex than the first-derivative methods, but they utilize the available information more fully and therefore converge more quickly. These methods involve setting up a system of simultaneous equations of size (3N — 6) (3N — 6) and solving for the atomic positions that are the solution of the system. Large matrices must be inverted as part of this approach. 

Another spectrophotometric method used the derivative method. Rajput and Raj [19] developed a first-order derivative zero-crossing method to analyze ezetimibe in combination with simvastatin. This method was further applied to determine ezetimibe in combination with lovastatin. The first derivation method also applied to determine ezetimibe in combination with rosuvastatin [20]. Besides the first-derivative method, second- and third-derivative methods were also reported for defermining ezetimibe as a single compound in its dosage form [21]. Flowever, the third-derivative method yielded the lowest limits of defecfion and quantitation relative to other methods used in this research. The maximum wavelength also remained constant regardless of fhe derivative method applied. 

A second approach is to calculate the change in potential-per-unit change in volume in reagent (AE/AV). By inspection, the endpoint can be located from the inflection point of the titration curve. This is the point that corresponds to the maximum rate of change of cell emf per unit volume of titrant added (usually 0.05 or 0.1 mL). The first-derivative method is based on the sigmoid shaped curve. 

The second-derivative method is an extension of the first-derivative method. The second-derivative of the data changes sign at the point of inflection in the titration curve. This change is often used as the analytical signal in automatic titrators. 

Usually, p is chosen to be a number between 4 and 10. In this way the system moves in the best direction in a restricted subspace. For this subspace the second-derivative matrix is constructed by finite differences from the stored displacement and first-derivative vectors and the new positions are determined as in the Newton-Raphson method. This method is quite efficient in terms of the required computer time, and the matrix inversion is a very small fraction of the entire calculation. The adopted basis Newton-Raphson method is a combination of the best aspects of the first derivative methods, in terms of speed and storage requirements, and the more costly full Newton-Raphson technique, in terms of introducing the most important second-de-... 

Tramadol and ibuprofen The first-derivative method was proposed for simultaneous determination of both compounds. The measurements of amplitude was done at 230.5 and 280 nm for tramadol (Trama) and ibuprofen (Ibu), respectively. The linearity was obeyed in the rage 5-50 pg mL-i for Trama and 5-100 pg mL-1 for Ibu. 29... 

Stability Region for the New Method with Vanished Phase-Lag and its First Derivative (Method NMI)... 

Figure 12.20 A typical redox titration curve for hydrogen peroxide measurement. The left ordinate is the millivolt reading from the redox electrode the right ordinate is the first derivative of the millivolt curve. Volume (mL) of titrant (permanganate) is shown on the abscissa. The endpoint shown is calculated with the first derivative method—the maximum of the pink curve. Used with permission from the author.

In Fig. 3 are shown the results obtained with a 1% alkali chitin solution. The cloud point can be determined as the temperature at which a very slight increment in the absorbance arises. From optical measurements it could be appreciated that the cloud point is centred at 30 C (Fig. 3a). The appearance of a slight opalescence in the solution at this temperature is in very close correspondence with the onset of changes in q equilibrium values. The latter are manifested by a well defined break-point in the Arrhenius-type plots (Fig. 3c). The dependence of the break-point values of each ti (T ) plot with frequency, CD, is also evident. The behaviour of the system was also monitored by the ratio of fluorescence intensities of pyrene at 384 and 372 nm, when excited at 343 nm. From the curve depicted in Fig. 3b, it can be appreciated a decrease in the fluorescence ratio when the system is heated with an inflexion point at 21.5°C (as determined by the first derivative method), signalling the onset of self-aggregation of the polymer. 

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