This exhaustive new E (x):
has the remarkable derivative:
Compare the two: we're within a whisker of a self-replicating function. The original and its derivative differ in only one term out of four. If I had started with a higher power than 4, giving a longer E (x), the difference in terms would be smaller yet: only one out of many.
The elegance of this E (x) is worth contemplating. I'll simplify it by dividing through by 4!:
Rearranged, this is:
This slight modification still gives an E (x) that differs from its derivative in only one term, the last: x4/4!. Conveniently, dividing by 4! has reduced that term's value — the match between E (x) and its derivative is now even closer.
Consider now that if I wrote an E (x) that started with the eighth power, it would differ from its derivative by only one term, x8/8!. Eight factorial is 40,320. For small values of x — say, less than 2 — that fraction is a very small amount indeed.
E (x) carried to the hundredth power would be identical to its derivative save for the amount of the last term, x100/100!. 100! is so vast that the fraction is exquisitely tiny even when x rises to 4 or 5.
The obvious question is, what would make that final error infinitely small — zero — regardless of the value of x? One obvious way...
... Is to let E (x) extend to infinite power.
As before, E (x) still equals its own derivative except for the last term. But now there is no last term; hence E (x) and its derivative are identical. (Notice that each term of E(x) is the derivative of the term to its right.) We have found the function we needed.
This is a remarkable, exotic function to achieve the simple property we were after. And yet each element of it is laid in place logically and without superfluity. It's a function of crystalline beauty: modular, structured, self-reflecting and never-ending.