What are $\partial/\partial f^j$ in Jost's definition of the differential mapping? Let $M$ be a $d$-manifold and $x_0=(x^1,x^2,\cdots, x^d)\in M$, Jost defines the tangent space at $x_0$ to be \begin{equation}\{x_0\}\times \operatorname{span}\left\{\frac{\partial}{\partial x^j}\right\}.\end{equation} So I understand these $\frac{\partial}{\partial x^j}$ to be symbols that specify the directions at $x_0$, and this seems to be a reasonable definition.
But then he defines the differential of $f:M\to N$, where $f=(f^i)$ is a differentiable function from $M$ to another manifold $N$ to be 
\begin{equation}v_j \frac{\partial}{\partial x^j}\mapsto v_j\frac{\partial f^i}{\partial x^j}\frac{\partial}{\partial f^i},\end{equation}and this is what got me confused.
What are these $\frac{\partial}{\partial f^i}$? If we just take $N=\mathbb{R}^c$, isn't the differential be the mapping \begin{equation}v_j\frac{\partial f^i}{\partial x^j}e_i\end{equation}where $e_i$ is the natural basis for $\mathbb{R}^c$?
So in my opinion these $e_i$'s should be independent of $f$ but in Jost's definition they seem to be related to the function $f$.
Where did I get it wrong? How do we actually understand these $\frac{\partial }{\partial f^i}$?
Thanks!
 A: Let me explain this as follows.
Suppose we have coordinates $(x^i)$ in a neighborhood of point $p$ and also coordinates $(y^{\lambda})$ in a neigborhood of point $f(p)$, so locally map $f$ can be identified with $y^{\lambda} = f^{\lambda}(x^i)$. As @LVK pointed the differential $Tf$ at $p$ is the linear map $T_{p}f \colon T_p M \to T_{f(p)} N$ that maps every coordinate vector field $\partial_i = \tfrac{\partial}{\partial x^i}|_{p}$ to $\frac{\partial f^{\lambda}}{\partial x^i} \frac{\partial}{\partial y^{\lambda}}|_{f(p)}$ and extended by linearity. These $\tfrac{\partial}{\partial f^{\lambda}}$ are just a shorthand notation for the fields $\frac{\partial}{\partial y^{\lambda}}|_{f(p)}$.
In fancier terms the differential of $f$ can be seen as a section of $T^* M \otimes f^* TN$ given by
$$
df = \frac{\partial f^\lambda }{\partial x^i} dx^i\otimes \frac{\partial}{\partial y^\lambda}
$$
where the Einstein summation is assumed.
The fields $\tfrac{\partial}{\partial f^{\lambda}} \Big|_{p} =  \left( \frac{\partial}{\partial y^{\lambda}} \right)_{f} \Big|_{p}= \frac{\partial}{\partial y^{\lambda}} \Big|_{f(p)}$ are a frame in the pullback bundle $f^* TN$
