I try do unterstand the connection between to different concepts:
1) The differential of a function $f:\mathcal{M}\to\mathcal{N}$ between two manifolds at a point $p\in\mathcal{M}$ is defined as $$\mathrm{d}_{p}f:T_{p}\mathcal{\mathcal{M}}\to T_{f(p)}\mathcal{\mathcal{N}}$$ $$v\mapsto(h\mapsto v(h\circ f))$$
or in other words: $[\mathrm{d}_{p}f(v)](h)=v(h\circ f)$ for $h\in C^{\infty}(N)$
2) The exterior derivative of a function $f:\mathcal{M}\to\mathbb{R}$ is a differential form of degree 1: $$\mathrm{d}f:\mathfrak{X}(\mathcal{M})\to C^{\infty}(\mathcal{M})$$ $$X\mapsto \mathrm{d}f(X):=X(f),$$ where we have identified a vector field $X$ as a derivation $X:C^{\infty}(\mathcal{M})\to C^{\infty}(\mathcal{M})$.
Now to my question....How are (1) and (2) connected with each other? I can write $$\mathrm{d}f(X)(p)=X(f)(p)= X_p(f)$$
where in the last step I have identified the vector field with the function $X:\mathcal{M}\to T\mathcal{M}$.
Is there a relation bewtween $\mathrm{d}_{p}f(X_{p})\in T_{f(p)}\mathbb{R}$ (which means: $\mathrm{d}_{p}f(X_{p}):C^{\infty}(\mathbb{R})\to \mathbb{R}$) and $X_p(f)\in\mathbb{R}$? It probably has to do something with the identification $T_{p}\mathbb{R}\cong\mathbb{R}$...
If I write momentarily $Df_p:T_pM\to T_{f(p)}\mathbb{R}$, I know that $Df_p(X_p)=a\frac{\partial}{\partial t}\big|_{f(p)}$ since $T_{f(p)}\mathbb{R}$ is spanned by $\frac{\partial}{\partial t}\big|_{f(p)}$ (where $t:\mathbb{R}\to\mathbb{R}$ is the standard coordinate on $\mathbb{R}$). To check who is this $a$, evaluate both sides on the function $t$: you get
$$Df_p(X_p)(t)=X_p(t\circ f)=X_p(f)=df_p(X_p)$$
from the left side and
$$a\frac{\partial}{\partial t}\Big|_{f(p)}t=a\frac{\partial t}{\partial t}\Big|_{f(p)}=a.$$
from the right side. So this rewrites finally as
$$Df_p(X_p)=df_p(X_p)\frac{\partial}{\partial t}\Big|_{f(p)}$$
giving you the wanted relation between the two differentials
$$df_p:T_pM\overset{Df_p}{\longrightarrow}T_{f(p)}\mathbb{R}\overset{\simeq}{\longrightarrow}\mathbb{R}$$
where $\simeq$ is the canonical identification $a\frac{\partial}{\partial t}\mapsto a$.