F[a_,b_]:= a  + b

D[a,c] == R1

D[F[a,b],c] = R1 + R2

Clear[a, b, f, r1, r2]
a /: Dt[a, c] := r1
b /: Dt[b, c] := r2
f[a_, b_] := a + b

Dt[f[a, b], c]
(* r1 + r2 *)

Clear[g]
Dt[g[a, b]]
(* Dt[b] D[g[a, b], b] + Dt[a] D[g[a, b], a] *)

Clear[a, b, f, r1, r2];

f[a_, b_] := a + b;

Dt[f[a, b]]

(* Dt[a, c] + Dt[b, c] *)

% /. {Dt[a, c] -> r1, Dt[b, c] -> r2}

(* r1 + r2 *)