X^M Derivative – Rules Of Calculus

l> Derivative x^n

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Derivative x^n
(wnyrails.org | Calculus | Derivatives | Table Of | x^n)

3 Proofs

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x^n = n x^(n-1)

Proof of x^n : from e^(n ln x) Given: e^x = e^x; ln(x) = 1/x; Chain Rule. Solve: x^n = e^(n ln x) = e^u (n ln x) (Set u = n ln x) = = x^n n/x = n x^(n-1) Q.E.D. Proof of x^n : from the Integral Given: x^n dx = x^(n+1)/(n+1) + c; Fundamental Theorem of Calculus. Solve: x^(n-1) dx = x^n / n x^n / n = x^(n-1) dx = x^(n-1) 1/n x^n = x^(n-1) x^n = n x^(n-1) QED Proof of x^n : algebraically Given: (a+b)^n = (n, 0) a^n b^0 + (n, 1) a^(n-1) b^1 + (n, 2) a^(n-2) b^2 + .. + (n, n) a^0 b^n Here (n,k) is the binary coefficient = n! / ( k! (n-k)! ) Solve: x^n = lim(d->0) ((x+d)^n – x^n)/d = lim < x^n + (n, 1) x^(n-1) d + (n, 2) x^(n-2) d^2 + ..
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+ x^0 d^n – x^n > / d = lim < (n,1) x^(n-1) d + (n, 2) x^(n-2) d^2 + .. + x^0 d^n > / d = lim (n,1) x^(n-1) + (n, 2) x^(n-2) d + (n, 3) x^(n-3) d^2 + .. + x^0 d^n = lim (n, 1) x^(n-1) (all terms on right cancel out because of the d factor) = lim (n, 1) x^(n-1) = n! / ( 1! (n-1)! ) x^(n-1) = n x^(n-1) QED

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