Experimenting about on this question, I some days ago investigated the polynomial equation:
$$p(x) = x^6 - d =0, p'(x) = 6x^5$$
I started with ${\bf X}_0{\bf = I}$
Then the iteration $$\begin{align}{\bf X}_{i+1} =& {\bf X}_i - p'({{\bf X}_i})^{-1} p({{\bf X}_i})=\\=& {\bf X}_i - {(6{{\bf X}_i}^5)^{-1}({{\bf X}_i}^6-{\bf D})}\end{align}$$
for $\bf D$ being matrix representation for the finite difference derivative filter [1,-1]/2. Converges in just 6 iterations (!) giving the following solution (and it's 6th integer powers):
We can see the 6 times exponents give the correct first, second and third discrete difference approximation of $\frac{\partial}{\partial x},\frac{\partial^2}{\partial x^2},\frac{\partial^3}{\partial x^3}$
What immediately springs to my mind is the typical long-tailed look indicative of something that may have something to do with a classical Infinite Impulse Response (IIR) filter in signal processing. But can we show / prove it somehow?
Here is an animation with power 32 instead of 6 illustrating all the way up to $X^{4\times 32}$ or in other words all the way from zeroth to fourth order derivative: $$\frac{\partial ^0}{\partial x^0},\frac{\partial ^1}{\partial x^1},\frac{\partial ^2}{\partial x^2},\frac{\partial ^3}{\partial x^3},\frac{\partial ^4}{\partial x^4}$$
