Reciprocal Rule to verify the Power Rule

I'm stuck on this one:

Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,

$\displaystyle \frac{d}{dx}(x^{-n}) = -nx^{-n-1}$

for all positive integers n.

----------------------------------------

Playing this out, I did this:

$\displaystyle \frac{d}{dx}(x^{-n}) = -nx^{-n-1} = -nx^{-(n+1)} = (-n)(\frac{1}{x^{n+1}})$

Next, I tried to restate the function like this:

$\displaystyle x^{-n} = \frac {1}{x^n}$

and then I apply the Reciprocal Rule:

$\displaystyle \frac{d}{dx}[\frac{1}{x^n}] = \frac{(x^n)'}{(x^n)^2} = \frac{nx^{n-1}}{x^{2n}}$

None of this is seems to be leading me anywhere helpful. Can anybody see where I might be going wrong or suggest another tack?

Thanks.