How to compute gradient of complicated scalar function ( limit and iteration)?

Question

I have a function which gives scalar potential:

$$P(c) = \lim_{n \to \infty} \frac{1}{2^n} \ln|f^{n}_c(0)|$$

where:

• $c$ is complex variable
• $f$ is the complex quadratic polynomial

$$f_c(z) = z^2 + c$$

• $f^n$ denotes iteration ( function composition )

$$f^{n+1} ~ \stackrel{\mathrm{def}}{=} ~ f \circ f^n$$

I would like to compute gradient of function P: $\nabla P $

I can approximate it using numerical methods:

Question: How can I compute gradient of function P: $\nabla P $ using symbolic methods ?

I have found only one description by Linas Vepstas

The gradient 2Df always points 'uphill':

$2Df = \frac{f z_n Dz_n }{ |z_n|^2 log |z_n|}$