In 1699, Newton formally introduced the first formulation of his technique for solving a polynomial equation in De analysi per aequationes numero terminorum infinitas. His original formulation was a purely algebraic procedure as calculus and the notion of the derivative had not yet been invented. Therefore, the original formulation differes in notation from the current iterative form stated explicitly in terms of the derivative. It is interesting that the correction term that Newton computed turned out to be the contemporary correction term which involves the derivative even though the derivative was not known at the time.
Secant Methods can be viewed as using a specific finite difference approximation to the derivative in Newton's Method instead of calculating it explicitly.