A Unified Geometric Framework for Higher Derivatives of Rational Functions

Computing higher-order derivatives of rational functions has classically relied on highly complex recursive formulations, such as repeated applications of the quotient rule or Faa di Bruno's formula. These traditional methods introduce severe computational bottlenecks, with algebraic complexity scaling quadratically as O(n²) with each successive differentiation step. This paper breaks this recursive bottleneck by introducing a novel, unified geometric framework that delivers an instantaneous, closed-form direct analytic generator of O(1) complexity By mapping the algebraic behavior of the function onto the static geometry of its underlying roots, we establish a rigorous mathematical link between the root discriminant (Δ) and the dynamic propagation of higher derivatives. We formulate and prove the Shifted Circle Theorem, demonstrating that under complex-conjugate regimes (Δ < 0), the derivatives are governed by periodic trigonometric trajectories, whereas under real-root regimes (Δ > 0), the system undergoes a topological transition into a hyperbolic subspace, collapsing the root geometry into real-valued exponential progressions along the horizontal axis Furthermore, we evaluate the boundary case of degenerate double roots (Δ = 0) using continuous limiting invariants via L-Hopital's rule to guarantee complete analytical closure across all structural transitions. We demonstrate that this closed-form solution achieves a direct cancellation of the factorial dynamic factor (n!) embedded in Taylor series kernels. This optimization enables the instantaneous generation of infinite power series expansions to arbitrary orders without requiring knowledge of antecedent derivatives. The theoretical robustness of the framework is validated through explicit numerical examples, highlighting its significant performance advantages for algorithmic implementations in computational physics, digital signal processing, and automated perturbation modeling