Reciprocal Rule to verify the Power Rule

**joatmon** · February 11th 2011, 09:25 AM

I'm stuck on this one:

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

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

for all positive integers n.
----------------------------------------
Playing this out, I did this:

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

Next, I tried to restate the function like this:

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

and then I apply the Reciprocal Rule:
$\frac{d}{dx}[\frac{1}{x^n}] = \frac{(x^n)'}{(x^n)^2} = \frac{nx^{n-1}}{x^{2n}}$

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.

**General** · February 11th 2011, 09:56 AM

You forgot a minus sign.

$\displaystyle \dfrac{d}{dx} \left( \dfrac{1}{x^n} \right) = - \dfrac{(x^n)^{'}}{(x^n)^2}=- \dfrac{nx^{n-1}}{x^{2n}}=-nx^{n-1-2n}=-nx^{-n-1}=-nx^{-(n+1)}=-n \, \dfrac{1}{x^{n+1}}$

**joatmon** · February 11th 2011, 10:08 AM

Thanks. I forgot that the quotient rule requires a negative in front of the fraction. That solves the whole thing.

Thread: Reciprocal Rule to verify the Power Rule

LinkBack

Thread Tools

Display

Reciprocal Rule to verify the Power Rule

Similar Math Help Forum Discussions

Power rule

Please verify my working for quotient and chain rule differentiation

Power rule.

Chain Rule, and Power Rule

Chain rule or Power rule or both? Trig derivative

Search Tags