Are there websites on which I can generate a sequence of functions?

Question

I'm trying to find a pattern on the following sequence of functions $g_n: (-1, \infty) \rightarrow \mathbb R$:

$$g_{n+1} (x) = g'_n(x) + g_n^2(x)$$

with $g_1(x) = \ln (1+x) + \frac x{1+x}$ and I would like to plot this sequence on some website/app/software. Wolframalpha is not that great when dealing with sequences of functions, do you know any?

Answer 1

If you know a little python you can find an online ide and whip up something pretty fast with sympy. Desmos also has the ability for recursive functions but it wouldn’t get you much information about the function.

Answer 2

With MATHEMATICA it suffices to write

g[1, x] = Log[x + 1] + x/(x + 1)
g[n_, x_] := D[g[n - 1, x], x] + g[n - 1, x]^2

g[3, x]

$$ -\frac{3}{(x+1)^2}+\frac{2 x}{(x+1)^3}+\left(\frac{2}{x+1}-\frac{x}{(x+1)^2}+\left(\frac{x}{x+1}+\log (x+1)\right)^2\right)^2+2 \left(\frac{2}{x+1}-\frac{x}{(x+1)^2}\right) \left(\frac{x}{x+1}+\log (x+1)\right) $$