f(x) = th((x-1)/(x+1)) = [sh((x-1)/(x+1))]/[ch ((x-1)/(x+1))]

Df: R/{-1}

f '(x) = [ch²((x-1)/(x+1)) ((x+1-x+1) /(x+1)²) - sh²((x-1)/(x+1)) . ((x+1-x+1) /(x+1)²)  ]/[ch²((x-1)/(x+1))]

f '(x) = [2/((x+1)².ch²((x-1)/(x+1)))].[ch²((x-1)/(x+1)) - sh²((x-1)/(x+1))]


f '(x) = 2/((x+1)².ch²((x-1)/(x+1)))

Et donc f '(x) > 0 pour x dans ]-oo ; -1[ U ]-1 ; oo[

->
f(x) est croissante sur ]-oo ; -1[
f(x) est croissante sur ]-1 ; oo[

lim(x-> -1-) f(x) = 1
lim(x-> -1+) f(x) = -1
-----
Sauf distraction.