We can also find this by using the difference quotient and the binomial theorem......suppose we want to find the derivative of xn ...... so we have.....
lim [ (x + Δx)n - xn ] / (Δx) Δx → 0
And by the binomial theorem, we can write
lim [ xn + C(n, 1)xn-1 Δx + C(n,2) xn-2 Δx2 + ..... +C(n, n-1) xΔxn-1 + C(n, n)Δxn - xn ] / (Δx)
Δx → 0
lim [C(n,1) xn-1 Δx + C(n,2)xn-2 Δx2 + ..... +C(n, n-1) xΔxn-1 +C(n, n) Δxn ] / (Δx)
Δx → 0
.... factor out Δx in the numerator...
lim (Δx) [C(n, 1) xn-1 +C(n, 2) xn-2 Δx + ..... +C(n, n-1) xΔxn-2 +C (n, n) Δxn-1 ] / (Δx)
Δx → 0
lim [C(n, 1) xn-1 + C(n, 2) xn-2 Δx + ..... +C(n, n-1) xΔxn-2 + C(n, n) Δxn-1 ]
Δx → 0
And taking the limit as Δx → 0 we have
C(n,1) x n-1 .......... but C(n, 1) is just n......so we have......
nxn-1
