I have a problem that relates to derivative of a delta function. The problem originates from a paper I was reading https://aip.scitation.org/doi/full/10.1063/1.2938860
In the paper, it is said that it is straightforward to see, as in equation (23), that:
$$\frac{\partial}{\partial \textbf{R}}\delta(\sum\limits_{i \in I} c_{I,i}\textbf{r}_i-\textbf{R}) = -\frac{1}{c_{I,k}}\frac{\partial}{\partial\textbf{r}_k} \delta(\sum\limits_{i \in I} c_{I,i}\textbf{r}_i-\textbf{R})$$
where $k$ can be any of $i \in I$, and I denotes a group of vectors $\textbf{r}_i$ for which $c_{I,i}\neq0$ for all $i$. $\textbf{R}$ is also a vector.
Is this correct? And how do you derive it in a simple way?
Thanks in advance.