I'm not going to write here the very old maths joke about ex and differentiation. It's not so funny, even if you would understand it. But I was wondering recently about all the beauty of this function and its differentiation and therefore also integration resistance.
Yes, in terms of a definition of differentiation, we will 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 is it built; for each independent variable x, which can be any real number - yes, there's no problem with zero or negative numbers, unlike when talking about logarithms (which makes sense as these guys are relatives, right? - they're their inverse functions... and thinking about the range of any basic non-translated exponential function, it makes sense when looking at the domain of its inverse - hello, logarithms)... So, for each such x we get a dependent value y aka f(x), in this case only 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 all this paragraph starts to be very messy and non-friendly for readers). As with all exponential functions (in their basic form), the graph is going through the point (0,1). It has positive (upward) sloping thanks to the base being higher than 1 (just to make things clear, e aka Euler's number equals circa 2.72, therefore >1). And yes, it looks like a curve starting somewhere on the left part of the number line very very near to the 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 way it goes to zero when talking about x approximating minus infinity, alright, 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 soon very high.
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, stay upside down on your head, but this function still stays the same.
I need to do some 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...
