How do you use the formal definition of differentiation as a limit to find the derivative of #f(x)=1/(x-1)#?

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Answer 1 · 2015-05-19T13:45:56

Have a look:
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Answer 2 · 2015-05-19T13:49:32

Firstly, let's remember the limit definition of the derivative :

#f'(x)=lim_(h->0)(f(x+h)-f(x))/h#

Here, we have the function #f(x) = 1/(x-1)#, so :

#f'(x)=lim_(h->0)(1/(x+h+1) - 1/(x+1))/h =lim_(h->0)(((x+1)-(x+h+1))/((x+h+1)(x+1)))/h#

#= lim_(h->0)((x+1)-(x+h+1))/(h(x+h+1)(x+1))#

#= lim_(h->0)((-h))/(h(x+h+1)(x+1))#

#= lim_(h->0)-1/((x+h+1)(x+1))#

#= -1/((x+0+1)(x+1)) = -1/(x+1)^2#.

That's it.