Defition of the derivative

(  [ (x + Δh) + 1 ] / [ (x + Δh) - 1 ]  -  [x + 1] / [ x - 1] ) / Δh  

{ get a common denominator }

(  [ (x + Δh) + 1 ]   [ x - 1]  -   [x + 1]  [ (x + Δh) - 1 ] ) /  ( [ Δh  [ (x + Δh) - 1 ]  [ x - 1] )  

{Simplify}

( x^2 + x Δh + x - x -  Δh - 1  -  x^2 -  xΔh + x - x -  Δh + 1 )  / ( [ Δh  [ (x + Δh) - 1 ]  [ x - 1] )    

( -2Δh)  / ( [ Δh  [ (x + Δh) - 1 ]  [ x - 1] )  

{ divide out   Δh  )

-2  / (   [ (x + Δh) - 1 ]  [ x - 1] )

Let  Δh → 0   and we have that the derivative is

-2  / ( x - 1)^2

WolframAlpha verifies this : 

https://www.wolframalpha.com/input/?i=derivative+%5B+x+%2B+1%5D+%2F+%5B+x+-+1%5D