In the Challenge Solution's mathematical model, how much does Firm 1's profit change as the subsidy, s, increases?

In the Challenge Solution, assume two firms face an inverse demand function of p=a-bQ, and the marginal cost of each firm is m. A per-unit subsidy, s, given to both firms, reduces each firm's after-subsidy marginal cost to m-s. The best-response function for Firm 1 is q1=(a-m+s)/2b - 1/2q2 and the best-response function for Firm 2 is q2 = (a-m +s)/2b - 1/2q1

Solving, each firm maximizes profits by producing such that q = (a-m + s)/3b

A change in the subsidy changes each firm's profit by derivative of profit i / derivative w.r.t s of......

Please explain your solution in details. Thank you.