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f(x) &= 6x^5 - 15x^4 + 10x^3 & x &\in [0, 1]

\end{align*}[/tex] and I found that by recursively applying it, that is [itex]f(f(x))[/itex] etc, I can get new functions with the same s-like shape but steeper slope.

I was curious if there was a way to smoothly go from [itex]x[/itex] (ie a line) to [itex]f(x)[/itex] to [itex]f(f(x))[/itex] and beyond, and some Googling told me this is called fractional iteration.

However the Wikipedia page didn't leave me with enough clues for finding the appropriate series. In particular I'm struggling to see how to expand the higher-derivative terms in the Taylor expansion in step 4. Any help with this would be appreciated.

FWIW I did go to university and had a fair share of calculus and linear algebra, but it's some 7+ years ago and I'm rusty these days. Oh and this is purely for fun, not homework.