How is the directional derivative used to determine the tangent map?

In general, let $M$ be a smooth manifold modelled on a Banach space $X_1$, let $p \in M$. Then, a choice of chart $(U, \alpha)$ about $p$ allows us to identify the tangent space $T_pM$ with the underlying model space $X_1$, which means one can construct a linear isomorphism $\Phi_{\alpha,p}: T_pM \to X_1$ (in finite dimensions, this can be seen as the reason why the tangent space dimension and the manifold dimension are equal)

Next, suppose also that we have another smooth manifold $N$ modelled on a Banach space $X_2$, and a smooth map $F:M \to N$. Then, as above, a choice of chart $(W, \beta)$ about $F(p)$ allows us to construct a linear isomorphism $\Phi_{\beta, F(p)}: T_{F(p)}N \to X_2$.

Now, we employ a tool often used in linear algebra: we study the map $F_{*p}$ by studying the "isomorphism related" map $S := \Phi_{\beta,F(p)} \circ F_{*p} \circ \Phi_{\alpha,p}^{-1}: X_1 \to X_2$. To keep all these spaces and maps clear in your mind, consider the following diagram (which is commutative by definition of $S$):

$\require{AMScd}$ \begin{CD} T_pM @>{F_{*p}}>> T_{F(p)}N \\ @V{\Phi_{\alpha,p}}VV @VV{\Phi_{\beta, F(p)}}V \\ X_1 @>>{S}> X_2 \end{CD}

By unravelling the definitions of $F_{*p}$, it shouldn't be too hard to prove that \begin{equation} S = d(\beta \circ F \circ \alpha^{-1})_{\alpha(p)}. \end{equation} (here the notation $dG_{\xi}$ means the Frechet derivative of $G$ at $\xi$, where $G$ is a map between Banach Spaces). Hence, inverting the relation in the diagram, we can write \begin{align} F_{*p} &= \Phi_{\beta,F(p)}^{-1} \circ S \circ \Phi_{\alpha,p} \\ &= \Phi_{\beta,F(p)}^{-1} \circ d(\beta \circ F \circ \alpha^{-1})_{\alpha(p)} \circ \Phi_{\alpha,p} \end{align}

This idea should be familiar from linear algebra, where given a linear transformation between finite-dimensional spaces, a choice of basis on the domain and target space allow us to rephrase everything about the linear map in terms of its corresponding matrix relative to those bases.

Hence, if you want to study the linear map $F_{*p}$, you just choose charts in the domain and target space of $F$, and consider the derivative of the "chart-representative" map $\beta \circ F \circ \alpha^{-1}$. Usually, the spaces $X_1$ and $X_2$ will be "familiar" to us, whereas the tangent spaces will be "abstract" (their elements could be complicated objects like equivalence classes of curves, or derivations etc), hence we prefer to work with $S$ rather than $F_{*p}$ directly. So, any property of linear transformations which is preserved under isomorphism (such as rank) can be studied more easily via the "induced map" $S$ rather than the original map $F_{*p}$.

So far everything I've said holds for general smooth manifolds $M,N$. But, in the case where $M$ and $N$ are finite dimensional vector spaces (not necessarily cartesian), we can make another simplification. Since they are vector spaces, the manifold structure is the one obtained by the identity chart, so in the discussion above, we can replace $\alpha = \text{id}_M$ and $\beta = \text{id}_N$ (and suppress the points $p$, $F(p)$ for convenience). Hence, our commutative diagram becomes $\require{AMScd}$ \begin{CD} T_pM @>{F_{*p}}>> T_{F(p)}N \\ @V{\Phi_{\text{id}_M}}VV @VV{\Phi_{\text{id}_N}}V \\ M @>>{S = dF_p}> N \end{CD}

Hence, we have that \begin{equation} F_{*p} = \Phi_{\text{id}_N}^{-1} \circ dF_p \circ \Phi_{\text{id}_M} \end{equation}

Hence, to determine the map $F_{*p}$ and its properties, all you need to do is compute $dF_p$ and its properties, and then "rephrase" all of that information into the language of tangent vectors via the isomorphisms.

Now, we have reduced the study of $F_{*p}$ to the study of $dF_p$. Note that for any $\xi \in M$, we have (by the chain rule) \begin{equation} dF_p(\xi) = \dfrac{d}{dt}\bigg|_{t=0} F(p + t \xi). \end{equation}

So as a conclusion: the bare minimum you need to do to determine $F_{*p}$ is the following:

• Pick a convenient basis $\{\xi_1, \dots, \xi_n\}$ of $M$.
• For each $\xi_i$ in the basis, compute \begin{equation} dF_p(\xi_i) = \dfrac{d}{dt}\bigg|_{t=0} F(p + t \xi_i) \end{equation} Since $dF_p$ is linear, by knowing its values on a basis, you know it everywhere on $M$.
• Explicitly write down the isomorphisms $\Phi_{\text{id}_M}: T_pM \to M$, and likewise for $N$. (This of course depends on your actual construction of the tangent space; whether you constructed it as equivalence classes of curves/ derivations/ germs/ quotient by a certain ideal etc)
• Perform the compositions mentioned above to calculate $F_{*p}$.

Reference: Loomis and Sternberg Advanced Calculus, section $9.4$ in particular, which is about tangent spaces (and also some earlier chapters for linear algebra)