Can you prove the Power rule using Induction?

The only problem I see with the proof the power rule using induction is that it will only prove it for integers.

Base case: k = 1:

We know that:

d/dn n^1
= d/dn n
= 1

Then, by the Power Rule, d/dn n^1 = (1)n^(1 - 1) = n^0 = 1. So it is true for the base case.

Induction step.

We have:

n^(k + 1)
= n^k*n^1
= n*n^k

Then, by the Product Rule:

d/dn n*n^k
= n^k + nk*n^(k - 1)
= n^k + (nk*n^k)/n
= n^k + k*n^k
= (k + 1)n^k
= (k + 1)n^[(k + 1) - 1]

Since this is equalivant to k*n^(k - 1) except with k + 1 in k's spot, we have shown it to be true for the next integer.

I hope this helps!