Can someone help me complete this proof of the power ruel I discovered?

Well, discover is the wrong word, I'm sure it has existed before this. I guess what I'm trying to say is I thought of a proof on my own without help?

d/dx(x^n)

def of derivative: [f(x+h) - f(x)] / h as h approaches 0

[(x+h)^n - x^n] / h as h approaches 0

using binomal theorum, (x+h)^n = [n choose 0 x^n + n choose 1 * x^n-1 * h + n choose 2 * x^n-2 * h^2... - x^n] / h

if h approaches 0, all terms with an h go to 0, so only n choose 0 x^n and -x^n remain.

n choose 0 x^n - x^n / h as h approaches 0

n choose 0 = n! / 0!(n-0)! aka n! / (n-0)! aka 1

x^n - x^n / h as h approaches 0

0/h as h approaches 0

0

...Obviously I made a mistake somewhere here. I can't seem to find where though. Can someone help?