How to discover the exponential function

Another article on the "How to discover/guess/prove/..." series written for a high school audience. The basic idea is to find a function whose derivative is itself, and to find the power series which satisfies this. Then messing with it to guess it must be the exponential function. No proofs, in fact, it is outrageously un-rigourous.  I hope the editor allows it.

I try to include only the most beautiful items, and state facts which I feel every high school student should know, even if they doesn't appear formally in the syllabus. 

Update (Nov 2017). The article was published in the November issue of At Right Angles. A nice surprise was Shailesh Shirali's companion article which gives some graphical intuition to complement the algebraic computations in my article. Here is the link to a reprint. 

Abstract

If a function is such that its derivative is the function itself, then what would it be? Some interesting mathematical objects  appear while trying to answer this question, including a power series, the irrational number $e$ and the exponential function $e^x$. The article ends with a beautiful formula that  connects $e$, $\pi$, the complex number $i=\sqrt{-1}$, $1$ and $0$.

Update: 15/June/2017. I was wondering what happened to this article, and the editor said he had sent some comments from the referee which were yet to be incorporated. I resent the article after incorporating the referee's comments, and now this article is slated to appear in the November issue of At Right Angles. Time to think about the next article in the series.

Here is a link to the updated preprint. Please do give comments.