# Reciprocal Rule to verify the Power Rule

• Feb 11th 2011, 09:25 AM
joatmon
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.
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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.
• Feb 11th 2011, 09:56 AM
General
You forgot a minus sign.

$\displaystyle \displaystyle \dfrac{d}{dx} \left( \dfrac{1}{x^n} \right) = - \dfrac{(x^n)^{'}}{(x^n)^2}=- \dfrac{nx^{n-1}}{x^{2n}}=-nx^{n-1-2n}=-nx^{-n-1}=-nx^{-(n+1)}=-n \, \dfrac{1}{x^{n+1}}$
• Feb 11th 2011, 10:08 AM
joatmon
Thanks. I forgot that the quotient rule requires a negative in front of the fraction. That solves the whole thing.