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I am trying to understand the basic rules of vector differentiation.

In a non-scalar expression where x is a column vector, is it valid to differentiate with respect to the kth element of x ?

For example, what would be the result of the following?

$ \partial/\partial x_k (x^T) $

$ \partial/\partial x_k (x) $

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Indeed, if you consider a vector $x$ to be a function of its entries. Then we have $$\frac{\partial}{\partial x_k}x = \begin{pmatrix} 0\\ \vdots\\ 1\\ 0\\ \vdots\\ 0\end{pmatrix} \text{ and }\frac{\partial}{\partial x_k}x^T = \begin{pmatrix} 0, \ldots, 1, 0, \ldots,0\end{pmatrix}$$ where the $1$ is in the $k^{th}$ spot. More generally, there is a notion of derivative for functions between any two Banach spaces (which includes $\mathbb R^n$ for all $n$), under which the derivative of a function $f$ at a point $a$ is the unique linear function $A$ such that $$\lim\limits_{x\to a} \frac{\|f(x)-f(a)-A(x-a)\|}{\|x-a\|}=0$$ and this agrees with the notion of a derivative from Calculus I because when $f$ is a function from $\mathbb R$ to $\mathbb R$ we get a linear function $cx$, and the constant $c$ is precisely the derivative (in the Calc I sense) of $f$ at $a$.

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