calculus and analysis – 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,