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

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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.

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