Let's say that I want to perform the functional derivative of a scalar functional $F$, for example
$F_{d}=\int d\mathbf{r}\left(\frac{\alpha(\rho)}{2}|\mathbf{p}|^{2}+\frac{\beta}{4}|\mathbf{p}|^{4}+\frac{K}{2}(\partial_{\alpha}p_{\beta})(\partial_{\beta}p_{\alpha})-v_{1}\frac{\rho-\rho_{0}}{\rho_{0}}\nabla\cdot\mathbf{p}+\frac{\lambda}{2}|\mathbf{p}|^{2}\nabla\cdot\mathbf{p}\right)$
but the derivative is taken in respect to a vectorial function $\bf p$:
$$\mathbf{h}=-\frac{\delta F_{d}}{\delta\mathbf{p}}=-\frac{\partial f}{\partial \bf p}+\nabla \cdot\frac{\partial f}{\partial\nabla \bf p}$$
$$h_i=-\frac{\delta F}{\delta p_{i}}=-\frac{\partial f}{\partial p_{i}}+\partial_{j}\frac{\partial f}{\partial\partial_{j}p_{i}}$$
where I just applied the definition [here]
So, my problem is, if in $f$ I have a term $f_1=\nabla\cdot\bf p$, what is the expression: $$\frac{\partial \nabla\cdot\bf p}{\partial\nabla \bf p}$$
and what happens for terms as $f_2=|\nabla\times\bf p|^2$?