Unlike the partial derivative, the or (total) differential (sometimes also called the total derivative) \(df(\mathbf{p})\) of a function \(f\) at point \(\mathbf{p}\in U\) takes all variable dependencies of all \(x_j, j=1,\dots,n\), into account simultaneously. The total differential \(df(\mathbf{p})\) (for fixed \(\mathbf{p}\)) is a linear map, which gives the best linear approximation of the function \(f\) at its evaluation point \(\mathbf{p}\). For a given choice of coordinates \(x_k\), \(k=1,\dots,n\) in the domain \(U\) of \(f\), the total differential \(df\) can be expressed in these coordinates by the so called Jacobian matrix

\begin{equation}
[J] =
\begin{pmatrix}
\frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} & \dots & \frac{\partial f}{\partial x_n}
\end{pmatrix}
\end{equation}

The Jacobian matrix is a collection of all partial derivatives with respect to all the coordinates, arranged in a specific order. All the partial derivatives are to be evaluated at point \(\mathbf{p}\in U\). The total differential \(df\) can then be written as:

\begin{equation}
df=[J]d\mathbf{x}=\sum_{j=1}^n \frac{\partial f}{\partial x_j}dx_j
\end{equation}

where \(d\mathbf{x} = (dx_1, dx_2, \dots, dx_n)^T\). In this sense, using the above purist terminology distinction, \(df\) is the (total) differential, while \([J]\) is the total derivative.

The total differential \(df\) of a function and its Jacobian matrix \([J]\) represent the same thing in a certain way, but are conceptually not quite identical: they are related in the same way as the linear map \(\varphi:V\rightarrow W\) between two vector spaces \(V\) and \(W\) was related to its associated matrix \(\rho\) with respect to a choice of bases in \(V\) and \(W\) in our linear algebra primer (we encourage you to review this there). There we had a linear map \(\varphi: V\rightarrow W\), which we were able to represent by matrix multiplication with a matrix \(\rho\) (the multiplication with which was a function \(\RR^n\rightarrow\RR^m\), once a particular choice of ordered bases in the two vector spaces \(V\) and \(W\) was made (we are assuming here in the notation compared to the linear algebra primer that these vector spaces are defined over the field \(\RR\)).. We also proved a proposition how the matrix \(\rho\) changed (for the same unchanged linear map \(\varphi\)) and became a different matrix, which we denoted by \(\rho’\)), when a different choice of bases in \(V\) and \(W\) was made.

Same thing here. Here, the total differential \(df(\mathbf{p})\) (for fixed \(\mathbf{p}\)) is the abstract linear map and the Jacobian matrix \([J]\) (also evaluated at point \(\mathbf{p}\)) is the associated matrix given a choice of basis; they are related in the same way as linear map \(\varphi\) and its matrix \(\rho\) in the linear algebra primer. The basis with respect to which [J] was taken is the basis of coordinate infinitesimals \(dx_j\). 

It is easy to see that the Jacobian matrix \([J]\) will change, if we pick a different set of coordinates \(x_j\), \(j=1,\dots,n\), because then the partial derivatives with respect to these different coordinates will have different values, as the coordinates will run in different directions (but the function \(f\) its total differential \(df\), will not have changed just because of a change of coordinates).

As a side note, we also notice that \(df(\mathbf{p})\) is a linear functional, i.e. a linear map \(\RR^n\rightarrow\RR\) from the vector space into its field of scalars, since its associated matrix has only one row. It operates on tangent vectors. In the language of differential forms it is a 1-form.

We can relate this understanding of total differentials of a function to the total derivatives which you know from high school (a slightly different but related concept), where you generally associated a total derivative \(\frac{df}{dt}\) of a function \(f(\mathbf{x}(t))\) with respect to a single parameter \(t\) as taking into account parameter dependencies all the way through, using the chain rule of differential calculus. In the case of nested functions, the total derivatives become a successive application of the linear maps which are the total differentials of the individual functions involved. For instance, consider function \(f(\mathbf{x})\) and let all the \(x_j\) above depend on a variable \(t\). This defines a one-parameter curve, along which we can compute the total derivative of \(f\) with respect to variable \(t\). It can be written as and equates to:

\begin{equation}
\frac{df}{dt} = \sum_{j=1}^{n} \frac{\partial f}{\partial x_j}\frac{d x_j}{d t} 
\end{equation}

This expression can be written in terms of Jacobian matrices of the nested functions \(f\) and \(\mathbf{x}\) involved: \(\frac{df}{dt}=[J]\dot{\mathbf{x}}\). The above expression is nothing but the Jacobian matrix of \(f:\RR^n\rightarrow\RR, \mathbf{x}\mapsto f(\mathbf{x})\), multiplied with the Jacobian matrix of the vector-valued function \(\mathbf{x}:\RR\rightarrow\RR^n, t\mapsto \mathbf{x}(t)\), which happens to be the tangent vector of the curve

\begin{equation}
\dot{\mathbf{x}}=
\begin{pmatrix}
\frac{d x_1}{dt} \\
\frac{d x_2}{dt}\\
\dots\\
\frac{d x_n}{dt}
\end{pmatrix}
\end{equation}

We see that in order to understand this properly, we must first define the total derivative and Jacobian matrix for vector-valued functions. We shall do this next. We also note from the above that differentials operate as linear maps on tangent vectors, not on the points in \(\RR^n\) (points on the curve) itself. But the differential \(df\) is evaluated at a point on the curve (fixed \(\mathbf{p}\)), before it acts as a linear map on a tangent vector.

Notational side note: What do the \(dx_i\) do in \(df\)? Let \(\mathbf{e}_j\) be a unit basis vector of tangent space associated with (in direction of) coordinate \(x_j\). Then \(dx_i(\mathbf{e}_j)=1\) if \(i=j\), and 0 otherwise. Working with matrices in coordinates (i.e. with \([J]\) instead of \(df\)), one does not write the \(dx_i\) and \(\mathbf{e}_j\) explicitly; they are implied by the position of the entry in the matrix/vector.