UPDATE!!! READ BELOW
Okay so we know that is the first derivative of , which can be also expressed as . And we know that if we take another derivative, it'll become , or .
When we integrate, we ask for the function whose derivative is the integral's argument function. So is , or (let's ignore for now).
Thus, we can say that is the antiderivative of , or its integral. With this in mind,
What is ?
And how about ?
Okay, so let's see. Well first of all, what would that even look like? Would you draw half a dash for the prime above ? Would you draw half an integral for ? Which half, the top half or the bottom half? Well, I think that's the least of the issues.
Here's what I propose:
If we're to stick to basic calculus and only use derivatives and antiderivatives, then, in other words, we're only using integers. So we're trying to get a rational number by using only integer sum and difference. How are we going to do that? We're going to cheat:
Consider the following:
And yet, I still don't know what would be (because I can't perform an infinite differentigration). What would it even mean?
Oh, and good luck on .
WHAT YOU GET WHEN YOU FIGURE THIS OUT:
[I'll tell you what the question mark is.]
UPDATE: The issue is resolved! View Fractional Calculus
Link Idea Credit: Michael Mendrin
Link 2: Differintegral (what I initially named this note, but then thought Derintegral was more brief. Should've left it ;))
Check out the cool concept application of Derintegrals:
Guess what this is?