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# Derivative

Robert P. Munafo, 2003 Sep 26.

The derivative of the Mandelbrot iteration function is taken with respect to C, and is computed as follows:

Z0 = 0
d/dCZ0 = 0

Z1 = Z02 + C = C
d/dCZ1 = 2 Z0 d/dCZ0 + 1 = 1

Z2 = Z12 + C = C2 + C
d/dCZ2 = 2 Z1 d/dCZ1 + 1 = 2 Z1 + 1 = 2 C + 1

Z3 = Z22 + C = C4 + 2 C3 + C2 + C
d/dCZ3 = 2 Z2 d/dCZ2 + 1 = 2 Z1 + 1 = 4 C3 + 6 C2 + 2 C + 1

Z4 = Z32 + C
d/dCZ4 = 2 Z3 d/dCZ3 + 1

(etc.)

The derivative has many applications:

It is the main part of the formula for the distance estimator, the best way to show the filaments in images of the Mandelbrot set.

For a point in a period-n mu-atom, the nth derivative of Zn tells how close that point is to the nucleus or to the edge of the mu-atom.

Using Newton's Method, the derivative can be used to locate all the points of bifurcation (bond points) from a mu-atom to its children. Together with the mu-atom size formulas, this can be used to locate all of the descendants of any given mu-atom.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Mar 31. s.27