Let $M$ be a smooth Manifold.
Let $p \in M$.
Let $U$ be an open neighborhood of $p$.
We denote by $C^{\infty}(U)$ the set of real valued smooth functions on $U$.
Let $\Lambda_p = \bigcup C^{\infty}(U)$, where $U$ runs through all open neighborhoods of $p$.
Let $f, g \in \Lambda_p$.
Suppose $f \in C^{\infty}(U)$ and $g \in C^{\infty}(V)$.
If there exists an open neighborhood $W$ of $p$ such that $W \subset U \cap V$ and $f|W = g|W$,
we say $f$ and $g$ are equivalent.
This is an equivalence relation on $\Lambda_p$.
We denote by $\mathcal{O}_p$ the set of equivalence classes on $\Lambda_p$.
Clearly $\mathcal{O}_p$ is an $\mathbb{R}$-algebra.
Let $f \in C^{\infty}(U)$, where $U$ is an open neighborhodd of $p$. We denote by $[f]$ the equivalence class containing $f$.
A derivation of $\mathcal{O}_p$ is a linear map $D\colon \mathcal{O}_p \rightarrow \mathbb{R}$ such that
$D(fg) = D(f)g(p) + f(p)D(g)$ for $f, g \in \mathcal{O}_p$.
The set $T_p(M)$ of derivations of $\mathcal{O}_p$ is a vector space over $\mathbb{R}$ and is called the tangent space at $p$.
Let $\epsilon > 0$ be a positive real number.
We denote by $\Gamma_p(\epsilon)$ the set of smooth curves $\gamma \colon (-\epsilon,\epsilon) \rightarrow M$ such that $\gamma(0) = p$.
Let $\Gamma_p = \bigcup_{\epsilon>0} \Gamma_p(\epsilon)$.
Let $(U, \phi)$ be a chart such that $p \in U$.
Let $\gamma_1, \gamma_2 \in \Gamma_p$.
Then $\gamma_1$ and $\gamma_2$ are called equivalent at $0$ if $(\phi\circ\gamma_1)'(0) = (\phi\circ\gamma_2)'(0)$. This definition does not depend on the choice of the chart $(U, \phi)$.
This defines an equivalence relation on $\Gamma_p(M)$.
Let $S_p(M)$ be the set of equivalence classes on $\Gamma_p(M)$.
For $\gamma \in \Gamma_p(M)$, we denote by $[\gamma]$ the equivalence class containing $\gamma$.
We will define a map $\Phi\colon S_p(M) \rightarrow T_p(M)$.
Let $c \in S_p(M)$.
Choose $\gamma \in \Gamma_p(M)$ such that $c = [\gamma]$.
Let $f \in C^{\infty}(U)$, where $U$ is an open neighborhood of $p$.
We write $D_c([f]) = (f\circ\gamma)'(0)$ for $f \in C^{\infty}(U)$.
Clearly $D_c$ is well defined and does not depend on the choice of $\gamma$.
Clearly $D_c \in T_p(M)$.
Hence we get a map $\Phi\colon S_p(M) \rightarrow T_p(M)$ such that $\Phi(c) = D_c$.
We claim that $\Phi$ is bijective.
Let $c, e \in S_p(M)$.
Suppose $D_c = D_e$.
Suppose $c = [\gamma]$ and $e = [\lambda]$.
Let $(U, \phi)$ be a chart such that $p \in U$.
Let $\pi_i:\mathbb{R}^n \rightarrow \mathbb{R}$ be the $i$-th projection map: $\pi_i(x_1,\dots,x_n) = x_i$.
We denote by $\phi^i$ by $\pi_i\circ\phi$.
Since $D_c([\phi^i]) = D_e([\phi^i])$, $(\phi^i\circ\gamma)'(0) = (\phi^i\circ\lambda)'(0)$.
Hence $(\phi\circ\gamma)'(0) = (\phi\circ\lambda)'(0)$.
Hence $\gamma$ and $\lambda$ is equivalent.
Thus $\Phi$ is injective.
Let $D \in T_p(M)$.
Let $(U, \phi)$ be a chart such that $p \in U$.
We assume that $\phi(p) = 0$.
We define $\phi^i$ for $i = 1,\dots,n$ as above.
Let $D([\phi^i]) = a_i$ for $i = 1,\dots,n$.
There exists $\epsilon > 0$ such that $(a_1t,\dots,a_nt) \in \phi(U)$ for every $t \in (-\epsilon, \epsilon)$.
Let $\gamma(t) = \phi^{-1}(a_1t,\dots,a_nt)$ for $t \in (-\epsilon, \epsilon)$.
Then it's easy to see that $\Phi([\gamma]) = D$.
Hence $\Phi$ is surjective and we are done.
The book you're looking for is our friend @Wedhorn's Manifolds, Sheaves and Cohomology.
Wedhorn's background is algebraic geometry, a subject in which he has already written (with his colleague Görtz ) a quite popular book, and his background shows in the book I recommend.
In Chapter 4 manifolds are presented as suitable ringed spaces i.e. topological spaces endowed with a sheaf of rings (sheaves having been explained in chapter 3) and then the author introduces tangent spaces, Lie groups, bundles, torsors and cohomology.
This approach has been advocated since at least 50 years ago but Wedhorn's is one of the very rare books that consistently adopts the ringed space approach to manifolds.
The prerequisites in topology, categories, homological algebra and differential calculus are presented in appendices, so that the book is quite self-contained.
Apart from its elegance and efficiency the ringed space approach to manifolds allows for a smoother (!) introduction to the more dificult theory of schemes (or analytic spaces) and is thus also an excellent investment for ulterior study of more advanced material.
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