Continuing the thought from last week, I've done a little bit of research. Not much, as the ants in my room were bothering me last week way too much to do anything besides thinking how far they could spread. (The joys of living in rental places – luckily, joys of the past, adds Veronika from the future while proofreading her "baby English" ten million years later... and those are the joys of non-native English speakers!). 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 that the derivative of eˣ is equal to itself. 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 the definition of differentiation, but I didn't like it. Later, I found one of many proofs that I really liked.

The idea is: let eˣ be equal to y, so we have a substitution eˣ = y. If we take that equation and apply a logarithm to it, we'll get ln(eˣ) = ln(y).

If we know how logarithms work, we know that ln(eˣ) = 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 variable and y being an unknown function (we're dipping our toes into differential equations here), if we differentiate this equation with respect to x, we'll get:

1 = (dy/dx) * (1/y)

Let's multiply both sides by y which gives us:

y = (dy/dx) * 1

After re-substituting y = eˣ, we'll get eˣ = d(eˣ)/dx, better written as eˣ = (d/dx) * eˣ or also eˣ = (eˣ)'.

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 wasn't so bad before, and it was 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 a kind, one-way-only way out.)