multivariable calculus - The notation $frac{partial}{partial x}$

In Jost's Riemannian Geometry and Geometric Analysis (Sect. 1.2, Chap. 1), the tangent space at a point $x_0$ in $\mathbb{R}^d$ is defined as
$$T_{x_0}\mathbb R^d=\{x_0\}\times E$$
where $E$ is the vector space spanned by $\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^d}$. Then the books says: "Here, $\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^d}$ are the partial derivatives at the point $x_0$." This is where I get confused. They are the partial derivatives of what? The only partial derivative I know is that of a function, but no function is given here.

Sure, if one wants to argue that $\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^d}$ are just formal notations here that don't mean anything other than a formal basis of $E$, then I can accept that even though I have doubts. But then there comes something that confuses me even more. If $f:\mathbb R^d\to\mathbb R^c$ is a differentiable map, then the derivative of $f$ at $x_0$ is defined to be (Einstein convention is used below)
$$df(x_0):T_{x_0}\mathbb R^d\to T_{x_0}\mathbb{R}^c\\
\quad v^i\frac{\partial}{\partial x^i}\mapsto v^i\frac{\partial f^j}{\partial x^i}\frac{\partial}{\partial f^i}$$

So apparently $\frac{\partial}{\partial f^j}$ here depend on $f$ and are not arbitraily selected, so the notation cannot simply be a formal one, which brings me back to the original question: what does $\frac{\partial}{\partial x^i}$ and $\frac{\partial}{\partial f^j}$ mean?

Answer

The tangent space $T_pM$ can be seen as the space of local linear operators acting on the functions $f: M \rightarrow \mathbb R$. If you have vector $v\in T_pM$ you can define how it acts on a function:
$$ v(f) = \left.\frac{d f(\gamma_v(t))}{dt}\right|_{t=0} $$
where $\gamma_v$ is any curve on $M$ such that $\gamma_v(0) = p$ and $\frac{d\gamma_v}{dt}(0) = v$.

Given a coordinate system $(x_i)$ you can find that there exist vectors in $T_pM$ that act on functions exactly like the partial derivatives $\frac{\partial}{\partial x_i}$, that is $v_i(f) = \frac{\partial f}{\partial x_i}(p)$. They are therefore denoted $v_i = \frac{\partial}{\partial x_i}$. Such vectors form a basis of $T_pM$, so any vector can be written as $$ v = v^i \frac{\partial}{\partial x_i} $$

If the point of differentiation is obvious, the vetor can be denoted as $v_i=\left.\frac{\partial}{\partial x_i}\right|_p$. Sometimes $\frac{\partial}{\partial x_i}$ can also denote the whole vector field, defining a vector at every point of the manifold.