Calculus of a Single Variable, Chapter 2, 2.1, Section 2.1, Problem 21

the alternate method to find the derivative of the function is the limit form.
By limit process, the derivative of a function f(x) is :-
f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]
Now, the given function is :-
f(x) = 1/(x-1)
thus, f'(x) = lim h --> 0 [{(1/(x+h-1)) - (1/(x-1))}/h]
or, f'x) = lim h --> 0 [{(x-1) - (x+h-1)}/{h(x-1)*(x+h-1)}]
or, f'(x) = lim h --> 0 [-h/{h*(x-1)*(x+h-1)}] = -1/{(x-1)*(x+h-1)}
putting the value of h = 0 in the above expression we get
f'(x) = -1/{(x-1)^2)}