Working from the definition: Let $F:M \to N$ a smooth map between smooth manifolds , for each $p\in M$ the function $$ dF_p : T_pM \to T_{F(p)}N$$ is called the $\textit{differential of $F$ at $p$}$. Given $v \in T_pM$, we let $dF_p(v)$ be the derivation at $F(p)$ that acts on $f \in C^\infty(N)$ by the rule $$dF_p(v)(f) = v(f \circ F)$$.
I need help understanding this for $F = det : M_2(\mathbb{R}) \to \mathbb{R}$. I am unsure how to understand $v(f \circ F)$ - how can $v(f \circ F) \in T_p\mathbb{R}\cong \mathbb{R}$, if $v\in T_pM_2(\mathbb{R}) \cong M_2\mathbb{R}$?