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Does anyone have a different way of explaining derivation of polynoms of all real numbers

Here is proof for positive power n

http://www.khanacademy.org/video/proof--d-dx-x-n?playlist=Calculus&sort=2#qa [Broken]

but I cant find for all real numbers

I dont want of a form which uses log rule like this one

http://bildr.no/view/1026598

But the binomial theorem is based on taylor series. My attempt to prove taylor series relies on integration rule sof polynomial which relies on derivation rules of polynomials which is what I want to prove?:

http://bildr.no/view/1030479

because I am using log rule to prove something else and to prove it I need to have proof for derivation of polynomials! To make a longer story more short:)

Here is proof for positive power n

http://www.khanacademy.org/video/proof--d-dx-x-n?playlist=Calculus&sort=2#qa [Broken]

but I cant find for all real numbers

I dont want of a form which uses log rule like this one

http://bildr.no/view/1026598

But the binomial theorem is based on taylor series. My attempt to prove taylor series relies on integration rule sof polynomial which relies on derivation rules of polynomials which is what I want to prove?:

http://bildr.no/view/1030479

because I am using log rule to prove something else and to prove it I need to have proof for derivation of polynomials! To make a longer story more short:)

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