Consider the first two games given as examples of normal form games.

In Prisoner's Dilemma the only Nash equilibrium is for both players to play $D$ : it's apparent that, no matter what player $1$ plays, player $2$ does better playing $D$ , and vice-versa for $1$ .

Battle of the Sexes has three Nash equilibria. Both $(O,O)$ and $(F,F)$ are Nash equilibria, since it should be clear that if player $2$ expects player $1$ to play $O$ , player $2$ does best by playing $O$ , and vice-versa, while the same situation holds if player $2$ expects player $1$ to play $F$ . The third is a mixed equilibrium; player $1$ plays $O$ with $\frac{2}{3}$ probability and player $2$ plays $O$ with $\frac{1}{3}$ probability. We confirm that these are equilibria by testing the first derivatives (if $0$ then the strategy is either maximal or minimal). Technically we also need to check the second derivative to make sure that it is a maximum, but with simple games this is not really necessary.

Let player $1$ play $O$ with probability $p$ and player $2$ plays $O$ with probability $q$ .

And indeed the derivatives are $0$ at $p=\frac{2}{3}$ and $q=\frac{1}{3}$ .