I am not sure if derivatives is the right ananlogy
Ok
What I am observed is the following
for the example $y=x^2$
if 0<x<1, repeated application of y will bring the value close to zero which "slows down" the higher the no. of iterations
But for x>1 and x<-1, repeated application of y brings it towards +infinity and -infinity respectively
for x=0,1,-1, the values remained unchanged no matter how many times you apply y onto it
Putting the above results together, it seems there's a set of x that "slips away" from x=1 and x=-1 at increasing rate with the no. of iteration k, and then there's a…