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!