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 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.

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