Question

39-42 Differentiate
`f` and find the domain of
`f\text{.}`

39.
`f(x)=(x)/(1-\ln (x-1))`

Answer 1

`\text{Given that,}\\\\f(x) = \frac x{1- \ln (x-1)}\\\\\\f'(x)=\frac{(1- \ln(x-1))-x\cdot \left(0-\frac 1{x-1} \right)}{(1- \ln(x-1))^2}\\\\\\~~~~~~~~= ((x-1)(1- \ln(x-1))+x)/((x-1)(1-\ln(x-1))^2)\\\\\\~~~~~~~~= (x-x\ln(x-1)-1+\ln(x-1)+x)/((x-1)(1-\ln(x-1))^2)\\\\\\~~~~~~~~= (2x-1-(x-1)\ln(x-1))/((x-1)(1-\ln(x-1))^2)\\`

`\text{domain f(x):} ~\x > 1, x\\eq e+1 \`