Leibnizs Formula

A nonselective thermal cleavage occurred, for instance, in the pyrolytic process accompanying the recording of the DQ mass spectrum of a four-components random copolymer containing units of j8-hydroxyvalerate (referred to as A), units of )8-hydroxyheptanoate (referred to as B), )8-hydrox-ynonaoate (referred to as Q, and jS-hydroxyundecanoate (referred to as The authors applied chain statistics (Leibniz formula, Eq. 2.50) and found... 

Then, applying (—0/0 )" and using Leibniz formula for repeated differentiation of a product, we first obtain... 

Differentiating this equation (n -I- 1) times using Leibniz s formula (5.82), we obtain... 

Shortcut methods were discovered that could speed up this limit process for functions of certain types, including products, quotients, powers, and trigonometric functions. Many of these methods go back as far as Newton and Leibniz. Using these formulas allows one to avoid the more tedious limit calculations. For example, the derivative function of sine xis proven to be cosine x. If the slope of sine x is needed at x= 4, the answer is known to be cosine 4, and much time is saved. 

Numerical Kekule Valence Structures. Kekule valence structures, like Clar s valence structures, may be viewed as geometrical formulas because they are drawn. Similarly, the novel notation for benzenoid hydrocarbons using an inscribed Greek jt represents a pictorial, qualitative, non-numerical representation of such molecules. However, well-designed or selected notation may have advantages that initially one did not anticipate. This has been the case with the Leibniz notation for derivatives in mathematics, and with graphical codes of configuration of /7-alkanes in structural chemistry. We will show now that the just introduced qualitative Jt-nota-... 

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