$y'=\frac{\left( x-1 \right)'.\sqrt{{{x}^{2}}+1}-\left( x-1 \right)\left( \sqrt{{{x}^{2}}+1} \right)}{{{\left( \sqrt{{{x}^{2}}+1} \right)}^{2}}}$

$=\frac{\sqrt{{{x}^{2}}+1}-\left( x-1 \right).\frac{2x}{2\sqrt{{{x}^{2}}+1}}}{{{x}^{2}}+1}=\frac{\sqrt{{{x}^{2}}+1}-\frac{{{x}^{2}}-x}{\sqrt{{{x}^{2}}+1}}}{{{x}^{2}}+1}$

$=\frac{x+1}{\left( {{x}^{2}}+1 \right)\sqrt{{{x}^{2}}+1}}=\frac{1+x}{\sqrt{{{\left( {{x}^{2}}+1 \right)}^{3}}}}$