Continuing the thought from last week, I've done a little bit of research. Not much, as the ants in my room (joys of living in rental places - luckily joys of the past, adds Veronika from the future while proofreading ten million years later her baby English... and those are joys of non-native English foreigners)... the ants were bothering me last week way too much to do anything besides thinking how far they can spread. It's not a good excuse, but it's still an excuse.
So let's see the proof first.

I've been looking for proof of the derivative of e^x being equal to its own. This means that for every x value, the slope at that point is equal to the y value. Of course, it is, but the proof was the first thing I wanted to see. Not that I couldn't do it on my own by using a definition of differentiation. But I didn't like it. Later I found one of many proofs which I really liked.

The idea is: let e^x being equal to y, so we have a substitution e^x = y. If we take that equation and use a logarithm on it, we'll get ln(e^x) = ln(y).
If we know how logarithms work, we know that ln(e^x) = x. I'll come back to this later when talking about logarithms (as they're also awesome!). So now we have x = ln(y).
Thinking about x being an independent value and y being an unknown function, if we differentiate this equation with respect to x, we'll get 1 = (dy/dx) * (1/y). Let's multiply both sides by y -> y = (dy/dx) * 1. After the re-substitution of y = e^x, we'll get e^x = d(e^x)/dx, better written as e^x = (d/dx) * e^x or also e^x = (e^x)'.
Tadaa. Yes, this one is good.

But how is it possible? What does it mean? What's the use of it? Those are questions I'd like to think about, but with the ants all over my room, I think maths will go a bit aside for a moment =)
(I'm a biologist too, did you know? It was not so bad before and it's like watching a documentary... but nowadays it's just too much, there are too many of them. I need to do something about it. About the ants... Show them some kind, one-way only way out.)