rhoprime[c_] := 
      p (1 - b) 
        Log[((d + (1 - b) (1 - 2 c))^2 c (b - 
                    d + (1 - b) c))/((1 - 
                      d - (1 - b) (1 - 2 c))^2 (1 - 
                    c) (d - (1 - b) c))]

rhosecond[c_] := D[rhoprime[c], c]

rhosecond[c]

c = (b - d)/(2 b - 1)

Simplify[((1 - b) (1 - 
                c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + 
                d) (-(((-1 + b) c (b + (1 - b) c - 
                            d) ((((1 - b) (1 - 2 c) + d))^2))/((1 - 
                            c) (((1 - (1 - b) (1 - 2 c) - d))^2) (((-(1 - b) c + d))^2))) - ((4 (1 - b) c (b + (1 - b) c - d) ((1 - b) (1 - 2 c) + d))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d))) + (((1 - b) c ((((1 - b) (1 - 2 c) + d))^2))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d))) - ((4 (1 - b) c (b + (1 - b) c - d) ((((1 - b) (1 - 2 c) + d))^2))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^3) (-(1 - b) c + d))) + (((b + (1 - b) c - d) ((((1 - b) (1 - 2 c) + d))^2))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d))) + ((c (b + (1 - b) c - d) ((((1 - b) (1 - 2 c) + d))^2))/((((1 - c))^2) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d)))) p)/(c (b + (1 - b) c - 
                d) ((((1 - b) (1 - 2 c) + d))^2))]

c  = ((b - d) d)/((b - 1) (2 d - 1))

Simplify[((1 - b) (1 - 
                c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + 
                d) (-(((-1 + b) c (b + (1 - b) c - 
                            d) ((((1 - b) (1 - 2 c) + d))^2))/((1 - 
                            c) (((1 - (1 - b) (1 - 2 c) - d))^2) (((-(1 - b) c + d))^2))) - ((4 (1 - b) c (b + (1 - b) c - d) ((1 - b) (1 - 2 c) + d))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d))) + (((1 - b) c ((((1 - b) (1 - 2 c) + d))^2))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d))) - ((4 (1 - b) c (b + (1 - b) c - d) ((((1 - b) (1 - 2 c) + d))^2))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^3) (-(1 - b) c + d))) + (((b + (1 - b) c - d) ((((1 - b) (1 - 2 c) + d))^2))/((1 - c) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d))) + ((c (b + (1 - b) c - d) ((((1 - b) (1 - 2 c) + d))^2))/((((1 - c))^2) (((1 - (1 - b) (1 - 2 c) - d))^2) (-(1 - b) c + d)))) p)/(c (b + (1 - b) c - 
                d) ((((1 - b) (1 - 2 c) + d))^2))]