>>10866639>>10867099Anyway: the sine, cosine, and exponential functions, among many others, all fit the necessary criteria. So we get:
We can simplify this greatly. As we all know, the derivatives of cycle between , , , and . So we get:
and , so we get:
Or simply:
By this same sort of process, we get:
You'll also notice that if you take the derivative of the Taylor series, you get the Taylor series, whose derivative is the Taylor series...
Anyway, the Taylor series of the exponential function (which I shall have denoted by ) is much simpler.
Of course, , and , so we get:
