c_t = -\frac{1}{V_r} \, \frac{\partial V_r}{\partial p}  = - \frac{\phi}{V_f} \, \frac{\partial}{\partial p} \left( \frac{V_f}{\phi}  \right) 


= - \frac{\phi}{V_f} \,  \left( V_f \, \frac{\partial}{\partial p} \frac{1}{\phi}  + \frac{1}{\phi} \frac{\partial V_f}{\partial p}  \right) 


= - \frac{1}{\phi} \, \frac{\partial \phi}{\partial p} - \frac{1}{V_f} \, \frac{\partial V_f}{\partial p}   = c_r + c_f