This research introduces a novel computational method that reverses the traditional mathematical approach by using definite integrals to evaluate highly complex higher order derivatives of quadratic and transcendent functions. By employing continuous phase shifts and boundary inversions, the method completely eliminates alternating signs from the calculation steps. This framework significantly increases efficiency and drastically reduces long manual derivations into a single direct step, which has been validated through advanced applications including shifted domains, the Shifted Circle Theorem, and advanced Bessel dualities.
Rida jamal badawi Abu sokon (Fri,) studied this question.