Jacobian matrix and determinant in the context of "Total derivative"

The total derivative of a generally vector-valued function ${\textstyle f\left(\mathbf {x} \right)}$ with a vector ${\textstyle \mathbf {x} }$ is its Jacobian matrix, ${\textstyle \mathbf {J_{f}} }$, whose entries are first-order partial derivatives of each component of ${\textstyle f}$ with respect to each coordinate of ${\textstyle \mathbf {x} }$. If ${\textstyle \mathbf {x} }$ has a dependency to another vector, let say ${\displaystyle \mathbf {x} \left(\mathbf {u} \right)}$, then the total derivative can be expanded to a matrix multiplication ${\textstyle \mathbf {J_{f}} \cdot \mathbf {J_{\mathbf {x} }} }$, where ${\textstyle \mathbf {J_{\mathbf {x} }} }$ is the Jacobian matrix of ${\textstyle \mathbf {x} }$, consisting of first-order partial derivatives of each component of ${\textstyle \mathbf {x} }$ with respect to each coordinate of ${\textstyle \mathbf {u} }$. If ${\textstyle \mathbf {u} }$ also has a dependency, let say ${\textstyle \mathbf {u} \left(\mathbf {p} \right)}$, then further expansion is possible in a similar manner; ${\textstyle \mathbf {J_{f}} \cdot \mathbf {J_{\mathbf {x} }} \cdot \mathbf {J_{\mathbf {u} }} }$. As a simple case, when ${\textstyle \mathbf {x} =\mathbf {x} \left(t\right)}$, it becomes ${\textstyle \mathbf {J_{f}} \cdot {\frac {d\mathbf {x} }{dt}}}$. All these expressions of the total derivative give the same meaning; it is the slope at a given point.