I'm reading Do Carmo's Differential Forms and Applications (1st ed) and on page 6 he takes a differentiable map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, a point $p \in \mathbb{R}^{n}$ and a vector $v \in \mathbb{R}_{p}^{n}$ (where $\mathbb{R}_{p}^{n} = \{ q-p: q\in \mathbb{R}^{n} \}$ is the tangent space of $\mathbb{R}^{n}$ in $p$), and then defines something in terms of $df_{p}(v)$, where $df_{p}: \mathbb{R}_{p}^{n} \rightarrow \mathbb{R}_{f(p)}^{m}$ is "the differential of the map $f$ at $p$".
My questions are:
(i) What is the differential of the map $f$ at $p$?
(ii) How can I calculate it in some specific vector like $v$?