binomial proof of positive integer power rule

We will use the difference quotient in this proof of the power rule for positive integers. Let $f(x)=x^n$ for some integer $n\ge 0$. Then we have

$f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{{(x+h)}^n-x^n}{h}}.$

We can use the binomial theorem to expand the numerator

$f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{C_0^n{x}^0{h}^n+C_1^n{x}^1{h}^{n-1}+\mathrm{\cdots }+C_{n-1}^n{x}^{n-1}{h}^1+C_n^n{x}^n{h}^0-x^n}{h}}$

where $C_k^n=\frac{n!}{k!(n-k)!}$. We can now simplify the above

$f^{\prime }(x)$$=\underset{h\to 0}{lim}{\displaystyle \frac{h^n+nx{h}^{n-1}+\mathrm{\cdots }+n{x}^{n-1}h+x^n-x^n}{h}}$

$=\underset{h\to 0}{lim}(h^{n-1}+nx{h}^{n-2}+\mathrm{\cdots }+n{x}^{n-1})$

$=n{x}^{n-1}$

$=n{x}^{n-1}.$