Bài giải:

Chọn A.

(I) ${y}'={{\left( -{{(x-1)}^{3}} \right)}^{\prime }}=-3{{(x-1)}^{2}}\le 0,\,\forall x\in \mathbb{R}$

(II) ${y}'={{\left( \ln (x-1)-\frac{x}{x-1} \right)}^{\prime }}=\frac{x}{{{\left( x-1 \right)}^{2}}}>0,\forall x>1$

(III) ${y}'=\frac{1.\sqrt{{{x}^{2}}+1}-x.{{\left( \sqrt{{{x}^{2}}+1} \right)}^{\prime }}}{{{x}^{2}}+1}=\frac{\sqrt{{{x}^{2}}+1}-x.\left( \frac{x}{\sqrt{{{x}^{2}}+1}} \right)}{{{x}^{2}}+1}$$=\frac{1}{\left( {{x}^{2}}+1 \right)\sqrt{{{x}^{2}}+1}}>0,\forall x\in \mathbb{R}$