Relative rate of change of a function (Q1570279)

A new field of real numbers is constructed, using the supermultiplication operation \(\otimes\), having the properties (i) commutativity, (ii) associativity, (iii) existence of a neutral element \(\delta\) and (iv) distributivity of \(\otimes\) relative to the operation \(*\) of multiplication. If the function \(a\otimes b\) is continuous in each argument in the domain \(a> 0,b>0\) and properties (i)--(iv) are satisfied, then \(\otimes\) is uniquely defined (to within a constant) by \(a\otimes b= a^{\log_\delta b}\). In the field constructed, the relative rate of change of a function is equal to its derivative, and the geometrical sense of derivation is in the character of a tangent (supertangent) to the function.