X to the power (a)n(d) a definition

What’s the derivative of x^n? The power rule states that it’s nx^{n-1}; and proving it with the construction e^{ln x^n}, the chain rule, the derivatives of e^x and \ln x you’re home fast.
But now let’s do it with the definition of the derivative and the binomial theorem:

    \[\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\]

and

    \[(x+y)^n=\sum_{k=0}^n \binom{n}{k}x^{n-k}y^k\]

Look at the video below

or

    \[f'(x)=\lim_{h\rightarrow 0}\frac{(x+h)^n-x^n}{h}\]

    \[\lim_{h\rightarrow 0}\frac{\binom{n}{0}x^n h^0+\binom{n}{1}x^{n-1}h+\binom{n}{2}x^{n-2}h^2+....+\binom{n}{n}x^{n-n}h^n-x^n}{h}\]

    \[\lim_{h\rightarrow 0}\frac{x^n+nx^{n-1}h+\binom{n}{2}x^{n-2}h^2+....+h^n-x^n}{h}\]

    \[\lim_{h\rightarrow 0}\frac{nx^{n-1}h+\binom{n}{2}x^{n-2}h^2+....+h^n}{h}\]

    \[\lim_{h\rightarrow 0}nx^{n-1}+\tbinom{n}{2}x^{n-2}h+....+h^{n-1}\]

    \[f'(x)=nx^{n-1}\]