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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?

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    So $\alpha$ is a function of $x$? Or of $x$ and additional variables?2012-07-09
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    If $\alpha$ is real valued, then you are right.2012-07-09
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    I assume $\alpha(\cdot)$ is a function of $x$; but it is unclear whether $\partial\alpha/\partial x=2x_k$ holds just for one $k$ or for all $k$.2012-07-09
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    @ChristianBlatter here I mean for $\partial f / \partial x_k$ to hold for any $x_k$ in the vector, and also renamed it to $f$ since I now realize $\alpha$ by convention is a scalar constant instead of a function per se.2012-07-10

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