Differentiating Cauchy Viscous strain tensor with Cauchy strain tensor

Hello all,

I am trying to investigate(as a part of my internship work) 2-3 nonlinear material models of visoelasticity. As usual, free energy has been divided in to two parts, viscous energy function and elastic energy function. In one model, I have elastic energy as a function of strain invariants,viscous energy as function of C(Right Cauchy strain tensor) and C_v(Right Cauchy viscous strain tensor) and a evolution equation of C_v with respect to time. To find stress tensor(of some kind) I have to differentiate it with C or deformation gradient but, I am unable to differentiate C_v by C. Initially, I was assuming that \frac{dC_v}{dC} will be zero but later I realised it is wrong. I have attached a pdf containing my functions and all other information. 

P.S.- This is my first question(or blog) if you find something wrong in the way I have written question or the way I asked. Please notify me.