Question

How do you differentiate `f(x) = x/(1-ln(x-1))`?

Answer 1

`f'(x)=(1-ln(x-1)+x/(x-1))/(1-ln(x-1))^2`

Explanation:

differentiate using the `color(blue)"quotient rule"`

`"Given " f(x)=(g(x))/(h(x))" then"`

`f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"`

`"here " g(x)=xrArrg'(x)=1`

`"and " h(x)=1-ln(x-1)`

`color(orange)"Reminder " d/dx(ln(f(x)))=(f'(x))/((f(x))`

`rArrh'(x)=-(1)/(x-1)`

`rArrf'(x)=(1-ln(x-1)-x(- 1/(x-1)))/(1-ln(x-1))^2`

`color(white)(rArrf'(x))=(1-ln(x-1)+x/(x-1))/(1-ln(x-1))^2`