I'm not going to write here the very old maths joke about eˣ and differentiation. It's not that funny, even if you understood it. But I was recently wondering about the beauty of this function and its resistance to both differentiation and integration.

Yes, in terms of the definition of differentiation, we can easily show that eˣ stays eˣ, regardless of all the "horror" of advanced calculus magic.

So, what's the deal with this function?

(Just to be clear, we're now talking about a friendly function of one variable, built above the domain of real numbers, nothing fancy...)

It's easy to understand how it's built: for each independent variable x... (which can be any real number – yes, there's no problem with zero or negative numbers, unlike with logarithms, which makes sense as these two are relatives, right? – they're inverse functions...) ...I got off a tangent here... So, for each such x, we get a dependent value y (also known as f(x)), which in this case will only be positive numbers. (Here we are – that's why logarithms cannot be built above non-positive numbers. We're back again at inverse functions, and now this whole paragraph is starting to look very messy and unfriendly for readers!) As with all exponential functions (in their basic form), the graph goes through the point (0,1). It has a positive (upward) slope thanks to the base being greater than 1 (just to make things clear, e, also known as Euler's number, equals approximately 2.72, therefore >1). And yes, it looks like a curve starting somewhere on the left side of the number line very, very near to zero (but remember, it will not cross the x-axis, as no matter what, eˣ will not give us a negative result. In its limit, it goes to zero when x approaches minus infinity, but no negative functional value!). On the right side, it goes crazy very soon and turns, going straight up to the sky. Because yes, powers are powerful and will push the number very high, very soon.

Anyway, what makes this function so extraordinary that even mathematicians make jokes about it? Bad jokes, but still... You can do whatever you want to it, stand upside down on your head, but this function still stays the same.

I need to do some more research about it and come back to it later, as I'm starting to be really curious.

And no, I'm not going to tell the joke here. Google it if you really want to know it...

Actually...

...when I was revising this post (almost a decade after I wrote it), I decided I was going to sneak in one of such jokes here, after all. (There are many variations...)

You're welcome. Or not—depending on whether you understand calculus and can appreciate the finer things in maths (don't worry if not, everything can be learnt, and then the meme below will become funny, I promise).