exponent y=x^a sequences.

Question

While analyzing square and cube functions, i found the following:

for y=x^2

x=1, y=1
          +3
x=2, y=4      +2
          +5
x=3, y=9      +2
          +7
x=4, y=16     +2
          +9
x=5, y=25

increase of increase (well, how else should i say this) is +2.

What does this signify?

Same pattern for y=x^3, only number in question being +6 and appearing after another round of measuring increase:

x=1, y=1
          +7
x=2, y=8      +12
          +19     +6
x=3, y=27     +18
          +37     +6
x=4, y=64     +24
          +61
x=5, y=125

Answer 1

I think that you are just showing that, whatever $x$ could be, the $n^{th}$ derivative of $x^n$ is a constant $$\frac{d^2}{dx^2}(x^2)=2$$ $$\frac{d^3}{dx^3}(x^3)=6$$ This corresponds to the number of steps required to arrive to your constant term. With $x^4$, one more round would give you $24$ as constant. In fact $$\frac{d^n}{dx^n}(x^n)=n!$$ what you would get after $n$ rounds.