Implicit differentiation at a point

I have the following implicit equation, and I’d like to compute the derivative of p with respect to T at a specific point. In other words, if I were to create the Contourplot of this equation, with p as the y-axis, and T as the x-axis, what would be its slope a certain point. I’m using Dt[] for the derivative, but it seems to give another implicit function, whereas I’m looking a numeric value. Code is attached.

derivativ =   Dt[p == \[Phi]/R + ((1 - \[Phi]) \[Phi] (1 - \[Lambda]) \[Beta])/(((W - T p) R + \[Phi] T) R + \[Phi]*(1 - \[Phi]) T)/(R (\[Lambda]/((W - T p) R + \[Phi] T  ) + ((1 - \[Lambda]) \[Beta] R)/(((W - T p) R + \[Phi] T) R + \[Phi] (1 - \[Phi]) T))), T]  derivativ /. {p -> 0.3192789688874802, T -> 313.206} 

I get the following warning:

General::ivar: 313.206` is not a valid variable. 

The output is:

Dt[0.319279, 313.206] == -142495. (-7.33049*10^-13 (0.16 + 1.05263 (0.2 + 1.05263 (-0.319279 - 313.206 Dt[0.319279, 313.206]))) -  9.02573*10^-14 (0.2 + 1.05263 (-0.319279 - 313.206 Dt[0.319279, 313.206]))) - 1.16674*10^-7 (0.16 + 1.05263 (0.2 + 1.05263 (-0.319279 - 313.206 Dt[0.319279, 313.206]))) 

Is it possible to get a number for the derivative evaluated at {p -> 0.3192789688874802, T -> 313.206}? And what does that warning actually mean?

Thank you,