Several Differentiable Formulas of Special Functions

The notation and terminology used in this paper have been introduced in the following articles: [13], [15], [16], [1], [4], [10], [12], [3], [6], [9], [7], [8], [11], [17], [5], [14], and [2]. For simplicity, we follow the rules: x, a, b, c denote real numbers, n denotes a natural number, Z denotes an open subset of R, and f , f1, f2 denote partial functions from R to R. One can prove the following propositions: (1) Suppose Z ⊆ dom(log (e) · f) and for every x such that x ∈ Z holds f(x) = a + x and f(x) > 0. Then log (e) · f is differentiable on Z and for every x such that x ∈ Z holds (log (e) · f)′↾Z(x) = 1 a+x . (2) Suppose Z ⊆ dom(log (e) · f) and for every x such that x ∈ Z holds f(x) = x − a and f(x) > 0. Then log (e) · f is differentiable on Z and for every x such that x ∈ Z holds (log (e) · f)′↾Z(x) = 1 x−a .