[tex]\text{Given that,}\\\\f(x) = \dfrac x{1- \ln (x-1)}\\\\\\f'(x)=\dfrac{(1- \ln(x-1))-x\cdot \left(0-\dfrac 1{x-1} \right)}{(1- \ln(x-1))^2}\\\\\\~~~~~~~~= \dfrac{(x-1)(1- \ln(x-1))+x}{(x-1)(1-\ln(x-1))^2}\\\\\\~~~~~~~~= \dfrac{x-x\ln(x-1)-1+\ln(x-1)+x}{(x-1)(1-\ln(x-1))^2}\\\\\\~~~~~~~~= \dfrac{2x-1-(x-1)\ln(x-1)}{(x-1)(1-\ln(x-1))^2}\\[/tex]
[tex]\text{domain f(x):} ~\{x|x > 1, x\neq e+1 \}[/tex]
