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I am trying to find the derivative of $y= (1-x^{-1})^{-1}$ to make it simple I use the chain rule and

$u = 1-x^{-1}$

$y = u^{-1}$

$y' = -1u^{-2}u' = -1(1-x^{-1})^{-2}(x^{-2})$

The problem is that this is not what my book gets and I am wondering if there is some strange power rule I overlooked or some other way to do this problem that gives me a different answer, or maybe I just need to factor is out somehow.

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    Most likely, another case where the book gives a simplified answer: $-1(1+x^{-1})^{-2}x^{-2}= {-1\over (1-{1\over x})^2}\cdot {1\over x^2} = {-1\over x^2\cdot(1-{1\over x})^2} = {-1\over \bigl( x(1-{1\over x})\bigr)^2} = {-1\over (x-1 )^2} .$2012-05-05

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$y = \left( 1 - x^{-1} \right)^{-1} = \frac1{1 - \frac1x} = \frac{x}{x-1} = 1 + \frac1{x-1}$ Hence, $y' = - \frac1{(x-1)^2}$

To proceed through you method of using chain rule, setting $u = (1-x^{-1})$, we have that $y = u^{-1}$. Hence, $y' = -\frac{u'}{u^2}.$ Note that $u' = \frac1{x^2}.$ Hence, we get that $y' = - \frac{1/x^2}{(1-x^{-1})^2} = - \frac{1/x^2}{(1-1/x)^2} = - \frac{1/x^2}{(x-1)^2/x^2} = - \frac1{(x-1)^2}$

The error you made was that you did not write $u^{-2}$ correctly. $\displaystyle u^{-2} = \frac1{(1-x^{-1})^2}$

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    @Jordan $(1-x^{-1})^2 = (1-1/x)^2 = (x-1)^2/x^2.$ Hence, $\frac{1/x^2}{(x-1)^2/x^2} = \frac1{(x-1)^2}.$2012-05-05