TITLE: Is $e$ a coincidence?
QUESTION [26 upvotes]: $e$ has many definitions and properties. The one I'm most used to is 
$$\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n $$
If someone asked me (and I didn't know about $e$):

Is there a constant $c$ such that the equation $\frac{d}{dx}c^x=c^x $ is true for all $x$?

Then I'd likely answer that: 

I doubt it! That would be a crazy coincidence.

I'm curious, is it a coincidence that there is a constant that makes this true?

REPLY [2 votes]: You have several good answers (which I have upvoted). Here's a partial answer (similar to @florence 's but without the definition of the derivative) that addresses this question:

Is there a constant $c$ such that the equation $\frac{d}{dx}c^x=c^x$ is true for all
  $x$?

without getting to the definition of $e$.
If you sketch the graph of $y=2^x$ (or think about population doubling) you will note that its slope is roughly proportional to its value - but it's never as steep as it is tall. The proportionality constant is less than $1$. If you do the same for $y=3^x$ you'll see that it's always steeper than it is tall. That makes it at least plausible that there's a constant $c$ with value between $2$ and $3$ that satisfies the differential equation. The particular value of that constant may be "coincidence" but its existence isn't.