i was watching the 3blue1brown’s calculus series to start building intuition and in the geometric interpretation of the power rule of derivatives he shows that the expression (x + dx)^ (n) is equal to x^(n)+ n x^(n-1) dx + (multiples of dx^2), and it kinda puzzled me cause he didn’t explain where it came from and i didn’t see it before

so i started trying a bunch of stuff and after struggling for some time i realized that the conventional algebra i was trying to apply was kinda blinding me cause trying to do the usual simplifications for, say, (x+dx) (x+dx) (x+dx) doesn’t tell you a lot, like saying that (x+dx)(x+dx) = x^(2)+ 2xdx + dx^(2)

at some point, i tried to just not simplify anything, so when doing (x+dx)(x+dx)(x+dx) you get (x x + x dx + dx x + dx dx) (x+dx), and finally (x x x + x dx x + dx x x + … + dx dx dx), which i realized, it’s just all the possible combinations for x and dx for n=3!! so the number of terms was 2^(3) basically, and 2^(n) in general

so ofc, in the end you will have exactly 1 x^(n) cause there’s only one possible combination where everything is x, exactly n terms of the form x^(n-1) dx, which can be seen in the diagram

dx x x x

x dx x x

x x dx x

x x x dx

where there are n possible combinations where exactly one element is dx, and the rest are x so the form of the terms are like mentioned before! and ofc the rest of the combinations, 2^(n) - n - 1 will just be multiples of dx^(2)

aand idk almost anything about group theory but like, yk, expressions that are like the roots of a polynomial? permutations? this sounds like galois doing foreshadowing waow… so like, idk i just found this rly cool and it’s just baby steps but like, i still wanted to share :D