A-Differentiability over Associative Algebras

The unital associative algebra structure (Formula presented.) on (Formula presented.) allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the (Formula presented.) -calculus. Thus, we introduce (Formula presented.) -differentiability. Rules for (Formula presented.) -differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are (Formula presented.) -differentiable, and their (Formula presented.) -derivatives are the power series defined by their (Formula presented.) -derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For (Formula presented.), we obtain (Formula presented.), and for (Formula presented.), (Formula presented.). Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations. © 2025 by the authors.