In a 2-player Cournot Model, player 1's profit function is a function of Q1 (player 1's production) and Q2 (player 2's production), and vice versa for player 2's profit function. To solve the Cournot Model, we take the derivative of each profit function with respect to the player's production, set it equal to zero, and then solve for Q1 as a function of Q2 (for player 1) and do the same for player 2 (Q2 as a function of Q1).

However, when we take the derivative of player 1's profit function, Q2 is treated as a constant; it is not treated as a function of Q1 though player 2's output will be a function of Q1. The same is the case for player 2: when we take the derivative of player 2's profit function, Q1 is treated as a constant.

Why is this the case? Why do we not treat Q2 as a function of Q1 in Player 1's profit function and why do we not treat Q1 as a function of Q2 in player 2's profit function?