Root Geometry and Higher Derivatives of Rational Functions

Closed formulas for the n-th derivative of rational functions, obtained by writing each quadratic factor through its roots. Complex-conjugate roots lead to a trigonometric law and real roots to a hyperbolic law, and partial fractions extend the method to linear numerators, cubic denominators, and Taylor coefficients.

Encoding a quadratic by its roots

Chapter 4 asks for the $n$-th derivative of a rational function without differentiating $n$ times. The key step is to write a quadratic denominator through its roots. When the discriminant is negative, the roots are $re^{\pm i\theta}$ and

Partial fractions and the elementary rule $\frac{d^n}{dx^n}(x-a)^{-1}=(-1)^n n!\,(x-a)^{-n-1}$ give the derivative at once. Writing $x-re^{i\theta}$ in polar form, with modulus $\rho(x)=\sqrt{Q(x)}$ and an angle $\phi(x)$ defined by $\cos\phi=(x-r\cos\theta)/\rho$ and $\sin\phi=r\sin\theta/\rho$, turns the result into a single trigonometric expression:

For example, $1/(x^2-x+1)$ has $r=1$ and $\theta=\pi/3$. At $x=1$ one finds $\rho=1$ and $\phi=\pi/3$, so the formula gives $f'''(1)=6$, matching direct differentiation.

Repeated differentiation of a rational function produces numerators that grow quickly with the order, and algebraic slips compound at each step. A closed formula avoids that growth: evaluating $f^{(n)}(x)$ costs essentially the same for every $n$, and the shape of the answer, a decaying radial factor times an oscillating angular factor, is visible at once. The book uses this structure to discuss where the zeros of high-order derivatives fall.

Two regimes, chosen by the discriminant

When the discriminant is positive, the roots are real, $re^{\pm\eta}$, and hyperbolic functions replace trigonometric ones. At $x=0$ the result is especially compact: $f^{(n)}(0)=n!\,\sinh((n+1)\eta)/(r^{n+2}\sinh\eta)$. The book's summary is a classification: complex roots give a trigonometric law, real roots a hyperbolic law. A linear numerator $Ax+B$ adds a cosine (or hyperbolic cosine) term, a case the book states as results it calls the Universal Trigonometric and Universal Hyperbolic Derivative Theorems.

General rational functions

Polynomial division and partial fractions reduce any rational function with real coefficients to a polynomial part, powers of linear factors, and quadratic factors. Each piece has its own closed law, so the derivative of the whole is a finite sum of closed forms. The book carries this to cubic denominators with one real root and a complex pair, to Taylor coefficients generated directly from the closed formulas, and to an observation about polynomials: if $f=(x-r)g$, then $f'-g=(x-r)g'$, so the curves $y=f'$ and $y=g$ meet exactly at $x=r$ and at the critical points of $g$.

How it connects to the other areas

The polar and hyperbolic coordinates here are the same ones used in mathematical geometry, now attached to the roots of a denominator instead of to the point $t+ib$. The goal of computing derivatives without repetition is shared with higher-order derivatives, and closed Taylor coefficients are a natural input for the coefficient-based transform framework.