Let $f:\mathbf R^n\to \mathbf R$ be a smooth map. We can write $df:T\mathbf R^n\to \mathbf R$ neatly as $$ df = \sum_{i=1}^n(\partial f/\partial x_i) dx_i $$
For a function $f:M\to \mathbf R$ defined on a smooth manifold $M$, given local coordinates $(x_1, \ldots, x_n)$ about a point $p$, we can write
$$ df = \sum_{i=1}^n(\partial /\partial x_i)f\ dx_i $$
Is there a similar neat way of writing the differential of the map $f:\mathbf R^n\to \mathbf R^m$, and more generally of $f:M\to N$, where $M$ and $N$ are smooth manifolds?
You've given a coordinate representation of the linear map $df_p:T_pM\to T_{f(p)}R$. One can work with it as a row-vector or introduce the "gradient vector" or indeed write it in the basis $\{dx_i$}.
Generally, for any $df_p:T_pM\to T_{f(p)}N$, its coordinate representation is simply the Jacobian matrix which could too be written in the corresponding basis matrices but usually its action on a tangent $v\in T_pM$ is simply written as a matrix-vector product $Jv.$