I'd appreciate help simplifying the relationship $ \nabla\left[ \; \phi(\parallel \mathbf{x} - \mathbf{\xi}_i \parallel) \; \right] $
for $\mathbf{x}$ and $\mathbf{\xi}_i$ in $\mathcal{R}^n$. This is how far I've come (I'm not even sure if I'm on the right track)
Setting $\mathbf{u} = \parallel \mathbf{x} - \mathbf{\xi}_i \parallel$, so that $ \nabla[ \phi(\mathbf{u}) ] = \left( \frac{\partial \phi}{\partial u_1} , \cdots , \frac{\partial \phi}{\partial u_n} \right) $
but
$ \frac{\partial \phi}{\partial u_j} = \frac{\partial \phi}{\partial u_j} \frac{\partial u_j}{\partial \mathbf{x}} + \frac{\partial \phi}{\partial u_j} \frac{\partial u_j}{\partial \mathbf{\xi}_i} $
for $j = 1 , \cdots , n$
Note: this question is related to a previous one
Edit: Your answers are correct, and I will tag them as such, but they aren't the answers I was hoping for. In my previous question, I required help proving a relationship between involving $\mathbf{x}$ and $\mathbf{\xi}$, from page 14 of these lecture slides. What I am now trying to understand is why $\phi$ is differentiated with respect to $\xi$ in the first place i.e. $\frac{\partial \phi}{\partial \xi}$. The problem I'm working on is in the area of Hermite interpolation. For example, on page 4 (column 1) of the paper Hermite variational implicit surface reconstruction it is shown that
$ \frac{\partial}{\partial f} \mathbf{n}_i^T \nabla f(\mathbf{\xi}_i) = \mathbf{n}_i^T \nabla k(\mathbf{x} , \mathbf{\xi}_i) $
In the past I assumed that the components of the gradient of $\nabla f(\mathbf{\xi}_i)$ and $\nabla k(\mathbf{x} , \mathbf{\xi}_i)$ were differentials of $f(\mathbf{\xi}_i)$ and $k(\mathbf{x} , \mathbf{\xi}_i)$ with respect to $x_i$. However the lecture slides and the paper suggest that the terms of the gradient are differentials with respect to $\xi_i$. What I really would like to know is why.