Ineed this equation,
ln (2x-1)/(x-1)=t dx/dt= (x-1)(1-2x)

The correct answer for this question is this one:

Starting with ln[(2X - 1)/(X - 1)] = t, solve for X in terms of t: 

(2X - 1)/(X - 1) = e^t > 

2X - 1 = (X - 1)*e^t > 

2X - X*e^t = 1 - e^t > 

X*(2 - e^t) = 1 - e^t > 

X = (1 - e^t)/(2 - e^t) = (e^t - 1)/(e^t - 2). 

Now differentiate ln[(2X - 1)/(X - 1)] = ln(2X - 1) - ln(X - 1) = t implicitly: 

(2/(2X - 1))*dX/dt - (1/(X - 1))*dX/dt = 1

dX/dt*((2*(X - 1) - (2X - 1)) / ((2X - 1)(X - 1))) = 1

dX/dt*(-1) = (2X - 1)(X - 1)

dX/dt = (X - 1)(1 - 2X).
Hope this helps you answer your question.