# Math2.org Math Tables: Derivative of x^n

(Math)
 xn = n x(n-1)

## 3 Proofs

Proof of xn : from e(n ln x)

Given: ex = ex; ln(x) = 1/x; Chain Rule.
Solve:

xn = e(n ln x)
= eu (n ln x) (Set u = n ln x)
= [e(n ln x)] [n/x] = xn n/x = n x(n-1)     Q.E.D.

Proof of xn : from the Integral

Given: xn dx = x(n+1)/(n+1) + c; Fundamental Theorem of Calculus.
Solve:

x(n-1) dx = xn / n
xn / n = x(n-1) dx = x(n-1)
1/n xn = x(n-1)
xn = n x(n-1)     Q.E.D.

Proof of xn : algebraicaly

Given: (a+b)n = (n, 0) an b0 + (n, 1) a(n-1) b1 + (n, 2) a(n-2) b2 + .. + (n, n) a0 bn
Here (n,k) is the binary coefficient = n! / ( k! (n-k)! )

Solve:

xn = lim(d->0) ((x+d)n - xn)/d
= lim [ xn + (n, 1) x(n-1) d + (n, 2) x(n-2) d2 + .. + x0 dn - xn ] / d
= lim [ (n,1) x(n-1) d + (n, 2) x(n-2) d2 + .. + x0 dn ] / d
= lim (n,1) x(n-1) + (n, 2) x(n-2) d + (n, 3) x(n-3) d2 + .. + x0 dn
= 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)     Q.E.D.