My general question is on what to do with a scalar result from a partial derivative.
Suppose a column vector $x$, and a function $f$ which accepts $x$ and returns a scalar. (e.g. proposition 8). Then suppose that in the process of finding $\frac{\partial f }{\partial x} $, that you take the partial derivative with respect to $x_k$ to get the following, $\frac{\partial f }{\partial x_k} = 2x_k $
so then, what is the derivative with respect to the entire vector $x$ (keeping in mind this is a hypothetical example and not the same as prop. 8)? I am not sure whether the the following result would be correct, and whether the result should a row or column vector. $\frac{\partial f }{\partial x} = 2x^T $
What if the function $f$ here returns a vector or matrix, then is the answer any different here?