# 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:

Z_{0} = 0

^{d}/_{dC}Z_{0} = 0

Z_{1} = Z_{0}^{2} + C = C

^{d}/_{dC}Z_{1} = 2 Z_{0} ^{d}/_{dC}Z_{0} + 1 = 1

Z_{2} = Z_{1}^{2} + C = C^{2} + C

^{d}/_{dC}Z_{2} = 2 Z_{1} ^{d}/_{dC}Z_{1} + 1 = 2 Z_{1} + 1
= 2 C + 1

Z_{3} = Z_{2}^{2} + C = C^{4} + 2 C^{3} + C^{2} + C

^{d}/_{dC}Z_{3} = 2 Z_{2} ^{d}/_{dC}Z_{2} + 1 = 2 Z_{1} + 1
= 4 C^{3} + 6 C^{2} + 2 C + 1

Z_{4} = Z_{3}^{2} + C

^{d}/_{dC}Z_{4} = 2 Z_{3} ^{d}/_{dC}Z_{3} + 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 n^{th} derivative of
Z_{n} 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) 1987-2023.

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