There are some trivial functions that are their own derivative. The zero function, y = 0, has a derivative of 0 everywhere. That is:

This doesn't go very far in letting us represent complex numbers, though, so it's probably not the self-deriving function we want. We need something with more structure.

As a starting point, assume that the function we want — call it E(x) — is just x raised to some power. It's a good start because it already comes close to what we want. For an power of 4, say, the function would be:

And its derivative is:

Well, x⁴ is hardly equal to 4x³. But they have the same general form — they are both powers of x — and their exponents only differ by 1. There is a scheme to redress that difference.

The next step is a bit like throwing something into a test tube to see what happens. E(x) needs some sort of modification to bring it closer to its derivative. Among the choices, the most elegant is just to add that derivative back into the original function itself. That is, instead of just x⁴:

Now the derivative of this new E (x) is:

Look well: E (x) and its derivative now share one term, 4x³, completely unchanged. That's to be expected, of course; we put the function's own derivative into it. But that suggests a pattern to be exploited, and the next step to is build an E (x) that includes not only its first derivative, but the next and the next, as far as possible. Using the same rule of derivation over and over, then, extend E (x) from just x⁴ to:

Or in simpler notation: