(d/dx) x^n

(Math | Derivatives | TableOf | x^n)

(d/dx) x^n : From (d/dx) e^(n ln x)

Given : (d/dx) e^x = e^x. (d/dx) ln(x) = 1/x. Chain Rule.

(d/dx) x^n = (d/dx) e^(n ln x)
= (d/du) e^u (d/dx) (n ln x) (Set u = n ln x)
= [e^(n ln x)] [n/x] = x^n n/x = n x^(n-1)

(d/dx) x^n : From the Integral

Given : (integral)x^n dx = x^(n+1)/(n+1) + c. Fundamental Theorem of Calculus.

(integral)x^(n-1) dx = x^n/n
(d/dx) x^n/n = (d/dx)(integral)x^(n-1) dx = x^(n-1)
1/n (d/dx) x^n = x^(n-1)
(d/dx) x^n = n x^(n-1)

(d/dx) x^n : Algebraicaly

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 (a,b) is the binary coefficient = a! / ( b! (a-b)! )

(d/dx) x^n = lim(d->0) ((x+d)^n - x^n)/d
= [ x^n + (n,1) x^(n-1) d + (n,2) x^(n-2) d^2 + .. + x^0 d^n - x^n ] / d
= [ (n,1) x^(n-1) d + (n,2) x^(n-2) d^2 + .. + x^0 d^n ] / d
= (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)
= n! / ( 1! (n-1)! ) x^(n-1) = n x^(n-1)