What question did this study set out to answer?

The aim is to overcome the computational challenges in deriving higher-order derivatives of rational functions using a unified approach.

July 6, 2026Open Access

The Shifted Circle Theorem: A Unified Geometric Mapping for Higher-Order Derivatives of Rational Functions

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Introduces a novel framework to compute higher derivatives of rational functions, suggesting significant computational efficiencies.

The aim is to overcome the computational challenges in deriving higher-order derivatives of rational functions using a unified approach.
Introduced the Shifted Circle Theorem to establish a relationship between the root discriminant and higher derivatives.
Formulated a geometric mapping to achieve O(1) complexity for higher-order derivative calculation.
Evaluated cases of complex and real roots, including degenerate double roots using L'Hopital's rule.
Achieved a closed-form solution for derivatives, allowing instantaneous power series expansions.
Demonstrated performance improvements over traditional methods with explicit numerical examples.
Showed that derivatives under complex roots follow periodic trigonometric paths, while real roots lead to hyperbolic dynamics.

Rida jamal badawi Abu sokon (2026) studied this question.

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