I need to predict the location of the minima x' in a free energy function F(x) in a regular solution model.

I need to predict the location of the minima x' in a free energy function F(x) in a regular solution model.

How do I determine the derivative ∂ΔF/∂x where the minima of the derivative obey the equation "ln( x'/(1-x'))= -cAB(1-2x')".

I need to show that for a poorly soluble solute (i.e. x' <<1), the above equation can be approximated asx' ~ exp (-cAB).