is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...? Is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...? 
And why is the difference always 2?
I think there is a relationship, but I can't get how and why...

This result does not need calculus to be shown, just knowing how to expand $(a+b)^n$ for $n = 2$:
$(x+1) = x^2+2x+1$
Then subtract off $x^2$ (since that is the preceding square) and you get:
$x^2+2x+1-x^2 = 2x+1$
so each successive difference will increase by two (as it depends on $x$, which is incrementing each time)
