f(x)= 1/ (x-1)

By the definition f'(x) = lim[(f(x+dx) - f(x))/dx] as dx --> 0.

Heref(x+dx) - f(x) = 1/(x+dx-1) - 1/(x-1) = [(x-1) - (x+dx-1)]/[(x-1+dx)*(x-1)] = -dx/[(x-1+dx)*(x-1)].

Now / this by dx and obtain -1/[(x-1+dx)*(x-1)]. When dx --> 0, (x-1+dx) --> (x-1), so-1/[(x-1+dx)*(x-1)] --> -1/[(x-1)^2] (for any x except x=1, of course).

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