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.
