General Formulas in the Table of 𝒈̅ − 𝑫𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆𝒔 for 𝒈̅ − 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏

Abstract

The consistent system of pseudo-arithmetical operations and its generator 𝑔̅are applied to 𝑔̅ − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 that are derived as solutions of some functional equations. Some interesting classes of 𝑔̅ − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 are built and the 𝑔̅ − 𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 for real functions marks a new development over the years. But, 𝑔̅ − 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 opens a new perspective highlighting the fundamental properties of these 𝑔̅ − 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 functions. Based on the fundamental properties of these 𝑔̅ − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠, we have further outlined our study in verifying other properties for pseudo-linearity, pseudo-nonlinearity and generalization of the table of 𝑔̅ − 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 for these transformed functions, with some pseudo-derivative identities as Pseudo-Basic Properties/Pseudo-General Rules. The table of 𝑔̅ − 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 for the 𝑔̅ − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 is built as a first attempt and equipped with the Pseudo-Linearity, the Constant Pseudo-Term, the Pseudo-Product, the Pseudo-Quotient and the Pseudo-Chain Rule as 𝑔̅ − 𝑓𝑜𝑟𝑚𝑢𝑙𝑎𝑠 grouped into eight cases. Further, it will be completed for different cases, as more modified functions take part in pseudo-nonlinear combinations, Elementary Transcendental Functions or 𝑔̅ − 𝑡𝑎𝑏𝑙𝑒 for 𝑔̅ − 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙𝑠 of 𝑔̅ − 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 etc., showing once again the importance of generated Pseudo-Analysis with the broad field of its applications as a generalization of Classical Analysis.