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$ \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 $\xi_i$.

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In fact we have $\frac{\partial}{\partial x}f(x-\xi) = - \frac{\partial}{\partial \xi}f(x-\xi)$ for any function $f$, of which your formula is a special case.

This can be seen by applying the chain rule for differentiation:

$\frac{\partial}{\partial x}f(x-\xi) = f^\prime(x-\xi)\cdot\frac{\partial}{\partial x}(x-\xi) = f^\prime(x-\xi)$ whereas $\frac{\partial}{\partial \xi}f(x-\xi) = f^\prime(x-\xi)\cdot\frac{\partial}{\partial \xi}(x-\xi) = -f^\prime(x-\xi)$.

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    Now I get it. If $u = x - \xi$. You only have to compare or equate both expressions so that $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} = -\frac{\partial f}{\partial \xi}$2011-06-09