By first principles, the derivative is
[tex]\displaystyle\lim_{h\to0}\frac{(x+h)^n-x^n}h[/tex]
Use the binomial theorem to expand the numerator:
[tex](x+h)^n=\displaystyle\sum_{i=0}^n\binom nix^{n-i}h^i=\binom n0x^n+\binom n1x^{n-1}h+\cdots+\binom nnh^n[/tex]
[tex](x+h)^n=x^n+nx^{n-1}h+\dfrac{n(n-1)}2x^{n-2}h^2+\cdots+nxh^{n-1}+h^n[/tex]
where
[tex]\dbinom nk=\dfrac{n!}{k!(n-k)!}[/tex]
The first term is eliminated, and the limit is
[tex]\displaystyle\lim_{h\to0}\frac{nx^{n-1}h+\dfrac{n(n-1)}2x^{n-2}h^2+\cdots+nxh^{n-1}+h^n}h[/tex]
A power of [tex]h[/tex] in every term of the numerator cancels with [tex]h[/tex] in the denominator:
[tex]\displaystyle\lim_{h\to0}\left(nx^{n-1}+\dfrac{n(n-1)}2x^{n-2}h+\cdots+nxh^{n-2}+h^{n-1}\right)[/tex]
Finally, each term containing [tex]h[/tex] approaches 0 as [tex]h\to0[/tex], and the derivative is
[tex]y=x^n\implies y'=nx^{n-1}[/tex]
