Prove f'(x) = nx^(n-1), if f(x) = x^n and n = positive integers

Prove f'(x) = nx^(n-1), if f(x) = x^n and n = positive integers. To prove this, we will use the formula f'(x) = the limit of {[f(x + delta x)- f(x)]/(delta x)} as delta x approaches to zero. Substituting (x + delta x)^n and x^n respectively to the formula. Applying the binomial theorem to (x + delta x)^n, simplify it.

After the necessary simplification by breaking down each and every terms to individual fraction, apply the limit as delta x approaches to zero. After applying the said limit, the term nx^(n-1) remains. And by that, we've proven f'(x) = nx^(n-1).

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