I want to calculate the functional derivative $\frac{\delta \mathcal{J}}{\delta f}$ for the functional:
$\mathcal{J}=\int d\mathbf{r}' |\nabla f(\mathbf{r}')|^2$.
With: $|\nabla f(\mathbf{r}')|^2=\displaystyle\sum_{i=1}^3 \left(\frac{\partial f}{\partial x_i}\right)^2$.
I expect to obtain from this functional derivative a scalar function.
I have proceeded this way:
$\frac{\delta \mathcal{J}}{\delta f}=\frac{\delta}{\delta f}\int d\mathbf{r}' |\nabla f(\mathbf{r}')|^2=2\int d\mathbf{r}'|\nabla f(\mathbf{r}')| \frac{\delta \nabla f (\mathbf{r}')}{\delta f(\mathbf{r})}=2\int d\mathbf{r}'|\nabla f(\mathbf{r}')| \frac{\nabla \delta f (\mathbf{r}')}{\delta f(\mathbf{r})}$
I'm not sure on what to do with this term: $\frac{\nabla \delta f (\mathbf{r}')}{\delta f(\mathbf{r})}$.