1. As most of you probably know, tetration by super-powers of integers is defined by iteration of exponentiation, as such:

In the part that uses exponential iteration, is used directly both in the base and the power. Or inversely under the radical and in the radicand itself.

Now, in general, is a function considered to involve tetration if is in both the base and the degree, even if there might be more "stuff" other than itself? Or at least be similar to or connected to tetration?

For example, is tetration since it involves iteration of exponentiation. But how about . In general, the base and the degree are not equal, so would this be tetration or at least somehow connected to it? Or is it just normal exponentiation? It's a rather stupid question, I know.

2.

3. It might make it more clear to show how I arrived at this...

Upon looking at limit representations of in standard functions, like...

... And you probably get the idea (there's more, but TeX is tiring!)

Anyway, I was like "Wow, is even in tetration". But then I wondered "Wait, is this even tetration?"

So, really, I was going to ask if was connected to tetration because of the above functions, but first of all I'm not sure if that's tetration.

4. Originally Posted by brody
For example, is tetration since it involves iteration of exponentiation. But how about . In general, the base and the degree are not equal, so would this be tetration or at least somehow connected to it? Or is it just normal exponentiation? It's a rather stupid question, I know.
I think this would be one for Guitarist to answer.
However, I don't think any of the two examples above actually are tetrations; they are just simple exponantiations to a power of x, which happens to be variable, but there is no chaining of exponentiations involved.
So, the answer to your original question is no, these constructs are not tetrations.

5. @brody

Tetration has one simple definition:

and

The function which you mentioned in your first post is defined as:

and

Please do not confuse the two. The first one (f) is tetration, the second one (g) is not tetration. When I was first getting into this subject, I used to call (g) the "lesser tetration" but I have since called (g) "iterated powers", as can be found on the Wikipedia page (Tetration - Wikipedia, the free encyclopedia). Another difference between these two functions is that (f) is hotly debated on the Tetration Forum (Tetration Forum), because people are still trying to extend the function to real/complex n in a consistent and unique way. The other function (g) is easily extended to real/complex n as follows:

Since exponentiation is a function of two parameters, we must chose one to stay the same each time when we start iterating. When we chose a constant exponent (as with g), then we get iterated power functions. When we choose a constant base (as with f), then we get iterated exponential functions. The two processes make very different results.

is an exponential function
is a power function

I would be inclined to say that anything that is not exactly tetration, shouldn't be called tetration. The subject is confusing enough as it is, even with a clear and simple definition. We don't need more confusion. With that being said, I would call all of your examples "exponential" in nature, and most functions of the form usually end up displaying some kind of exponential behavior. So if you want to call these expressions something, then call them exponential expressions, or hyperarithmetic expressions, but I wouldn't call them tetration.

6. Originally Posted by andydude
@brody
Thanks a lot, andydude. Great reply, I must say. After I initiated this thread, I thought "Well, isn't exponentiation, there's nothing special about that, it's just multiplication. So shouldn't be either". I know tetration is strictly defined... maybe it's the non-commutativity, or the superscript notation that made those functions stand out. But then again, seeing the unique contrast between functions with a variable degree and functions with a variable base, then maybe there is something special to functions with both variable degrees and bases.

Either way, I now understand it can't be true tetration. And from my own conclusion, they aren't tetration, but they're interesting none-the-less.

Now, do you mind if I ask one more question? Can I apply the power rule of differentiation to iterated degrees?

According to Wolfram Alpha. I'm probably overlooking a simple mistake, but shouldn't it be ?

7. You seem to be confusing the total derivative (Total derivative - Wikipedia, the free encyclopedia) and the partial derivative. It's an easy mistake to make, since they make the same result for univariate functions. However, to differentiate we need to make it into a bivariate function, which requires the total derivative, not the partial derivative.

To use this total derivative we need to find:

,

,

So, putting this all together:

With this understanding, your previous guess was just , you forgot the part.

8. Why would you bi-variate the function? It seems fine without any multivariate methods. It's uni-variate, and I don't see any reason that it should change to the other form.

9. "It's uni-variate"

Seriously? OK, back to basics.

What is the chain rule? It's a rule that you use when you differentiate a composition of functions. Your function is a composition of functions, the only non-trivial function being exponentiation. Exponentiation is bivariate. Because exponentiation is bivariate, and (f) is a composition of trivial functions and exponentiation, we need to differentiate exponentiation as a bivariate function
in order to differentiate (f). So even though (f) is a univariate function, it is built with a bivariate function, so we need to treat it as such when we apply the chain rule.

If you really want to treat (f) as a univariate function without any compositions, then you have to go back to the limit definition of the derivative, because
does not satisfy any of the simple rules they give you in calculus (it is not a power function, and it is not an exponential function, so you cannot use either of those rules). This might be several pages of work, and in the end you will surely get the same answer as above, but if you really want to do it that way, you can.

10. Originally Posted by andydude
"It's uni-variate"

Seriously? OK, back to basics.
Sorry. Unfortunately, my calculus course hasn't gotten to this yet, it seems. I asked a reliable, more intelligent friend and he also thought one would use the differential power rule. But, of course I should have noticed that using that would've brought about the original function meaning it is its own derivative, which is obviously not true.

because does not satisfy any of the simple rules they give you in calculus (it is not a power function, and it is not an exponential function, so you cannot use either of those rules)
Ah, okay. I thought the rule could simply apply to for (which as I realized couldn't be true). Since the only difference is that the degree is equivalent to the base, you could still apply the rule. However not, as you explained QED.

Anyway, I'll try getting out my old texts and see if I could learn more from this. Thanks a lot for your help!

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