Conjecture + Calculus = Conjectulus?

Discrete Mathematics Level 4

Have you ever took some playing cards and start to make a pyramid of them, like a house of cards? Well, Mohamed has already played it a lot of times. Here's the image of what I'm talking about:


So, Mohamed was very interesting about the way that the pyramid behaviors, first because it kinda looks like a growing tree, and second weather was possible to predict how the structure would looks like in a certainly quantity of cards and "floors". And then he start to make his conjecture about the pyramid, ending with success. Here it is what he got:

A formula to predict how many \(\textbf{small towers}\) will have this pyramid;

A formula to predict how many \(\textbf{cards}\) will need to make the pyramid; and

A formula to predict how many \(\textbf{shapes}\) of \(\textbf{small triangles}\) will have the pyramid.

All three formulas which obey the proportion of "How many floors" has the pyramid.

After that, satisfied with his result, he come back to imagine the growing of the pyramid like a tree does, in a constant process. With that in mind, he calculated the instantaneous derivative of the three functions.

Answer this: what is the value of the sum of the three instantaneous derivative when the tree pyramid has 68 "floors"?

\(\textbf{Details and assumptions:}\)

Consider that all the cards are \(\textbf{perfect aligned}\) as I drew in the paint:

When calculating the derivative of the functions, consider to all real numbers.

*This is a original problem (I think...). In the case of ambiguity, just let me know so I can see if I can fix it. \(\textbf {I had to fix it two times}\).


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