Does:
Solve[Dt[x^2*Cos[y]^2 - Sin[y] == 0, x], Dt[y, x]]
(* {{Dt[y, x] -> (2 x Cos[y])/(1 + 2 x^2 Sin[y])}} *)
Solve[Dt[x^2*Cos[y]^2 - Sin[y] == 0, x, x], Dt[y, x, x]]
(* {{Dt[y, {x, 2}] -> (
Cos[y] (2 - 2 x^2 Dt[y, x]^2 - 8 x Dt[y, x] Tan[y] +
Dt[y, x]^2 Sec[y] Tan[y] + 2 x^2 Dt[y, x]^2 Tan[y]^2))/(
1 + 2 x^2 Sin[y])}} *)
suffice?