Correspondence To Analytic Derivative
When the ring R of scalars is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus. The element Y-X of the ring R divides Yn - Xn for any nonnegative integer n, and therefore divides f(Y) - f(X) for any polynomial f in one indeterminate. If we denote the quotient (in R) by g:
then it is not hard to verify that g(X,X) (in R) coincides with the formal derivative of f as it was defined above.
This formulation of the derivative works equally well for a formal power series, assuming only that the ring of scalars 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. This way differentiation becomes a part of algebra of functions.
Read more about this topic: Formal Derivative
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