Prove that \(\left.\dfrac{d^n}{dx^n}\right|_{x=0}x^n\times n^x=n!\) for all positive integer \(n.\)

Find, with proof, \(\dfrac{d^n}{dx^n}\dfrac{x^n}{1-x}.\)

Find, with proof, \(\dfrac{d^n}{dx^n}x^{n-1}\log x.\)

Prove that \(\left.\dfrac{d^n}{dx^n}\right|_{x=0}x^n\times n^x=n!\) for all positive integer \(n.\)

Find, with proof, \(\dfrac{d^n}{dx^n}\dfrac{x^n}{1-x}.\)

Find, with proof, \(\dfrac{d^n}{dx^n}x^{n-1}\log x.\)

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