Fix $N\in \mathbb{N}$. Let \begin{equation} H(\mathbb{w}) := \frac{1}{1+e^{-\sum_1^Nw_i}},\quad (\mathbb{w}\in\mathbb{R}^N). \end{equation} I am interested in the symmetries of its partial derivative (say, wrt $w_1$): \begin{equation} h(\mathbb{w}) := \frac{\partial}{\partial w_1} H(\mathbb{w}) = \frac{e^{-\sum_1^Nw_i}}{\left(1+e^{-\sum_1^Nw_i} \right)^2}. \end{equation} Specifically, I am interested in these symmetries when $N>1$. When $N=1$, I know that $h(\cdot)$ is symmetric and unimodal about $w=0$, i.e. $$ h(w) = h(-w) \ \forall \, w\in\mathbb{R}. $$
I am having trouble thinking about the analogous symmetries for $N\geq 2$, and would appreciate any help in this regard.