Why is the derivative of x^n, nx^(n-1)?
From the definition of a derivative: f'(x) = lim h->0 ((f(x+h) - f(x)) / h) Let f(x) = x^n --> d\dx x^n = lim h->0 (((x+h)^n - x^n) / h) By binomial expansion, (x+h)^n = x^n + nhx^(n-1) + n(n-1)h^2 x^(n-2) + ... + h^n --> d\dx x^n = lim h->0 ((x^n + nhx^(n-1) + n(n-1)h^2 x^(n-2) + ... + h^n - x^n) / h) = lim h->0 (nx^(n-1) + n(n-1)h x^(n-2) + ... + h^(n-1)) = nx^(n-1)
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