Consider the following function (all roman symbols are integers)
$$ f(x) = (x+1)(x+2)(x+3) $$ Then $$ f'(x) = (x+2)(x+3) + (x+1)(x+3) + (x+1)(x+2) $$ Now I want to find a way to write this in a sum/prod way and I thought of this: $$ f'(x) = \sum_{m \geq 1 }^{3} \, \prod_{i=m}^{(m+1)\text{mod}3} (x+i) $$ then for $m=1$ we get the product $(x+1)(x+2)$, for $m=2$ we get $(x+2)(x+3)$ and for $m=3$ we get the product $(x+3)(x + 4\text{mod}3) = (x+3)(x + 1)$ which gives buck what we originally wanted.
My question is a) is the notation ok? and b) wether it is safe to generalise this to the following: if $$ f(x)=\prod_{i=1}^{r}(x+i) $$ then $$ f'(x) = \sum_{m \geq 1}^{r}\prod_{i=m}^{(m+r-1)\text{mod}r}(x+i) $$ is the correct result?