New ideas to the design of algorithms based on derivatives

This article proposes new perspectives for developing derivative based numerical algorithms, supported by the introduction of a generalized derivative operators. It demonstrates that these operators have the potential to enhance and extend existing derivativebased numerical methods. To this end, two iterative derivative driven methods are examined and refined: the Newton Raphson method and the Gradient method. For both approaches, generalized derivatives are introduced with the goal of reducing the number of iterations required for convergence. These modifications are presented through geometric interpretations of the proposed constructions, which clearly illustrate their convergenceaccelerating properties. The concluding remarks emphasize the significant opportunity to advance and refine numerical algorithms through the use of generalized derivatives.