above. This formulation of the derivative works equally well for a formal power series, as long as the ring of coefficients is commutative. Actually, if the division in this definition is carried out in the class of functions of continuous at , it will recapture the classical definition of the derivative. If it is carried out in the class of functions continuous in both and , we get uniform differentiability, and our function will be continuously differentiable. Likewise, by choosing different classes of functions (say, the Lipschitz class), we get different flavors of differentiability. In this way, differentiation becomes a part of algebra of functions. See also Derivative Euclidean domain Module of relative differential forms Galois theory Formal power series Pincherle derivative References Michael Livshits, You could simplify calculus, arXiv:0905.3611v1 Abstract algebra