$ \frac{\partial \phi}{\partial \xi} ( \parallel \mathbf{x} - \xi_i \parallel) = - \frac{\partial \phi}{\partial x} ( \parallel \mathbf{x} - \xi_i \parallel) $
from page 14 of these lecture slides, where $\phi$ is (a radial basis) function the variables $\mathbf{x}$ and $\xi_i$.