I think I have this right, but I have no way to check it and would appreciate a second opinion.
I want to calculate the following:
$$ \frac{\partial}{\partial\rho_\nu}\left[\left(\partial_\mu\rho_\nu-\partial_\nu\rho_\mu\right)\left(\rho^\nu-\rho^\mu\right)\right] = \frac{\partial}{\partial\rho_\nu}\left(\partial_\mu\rho_\nu\rho^\nu-\partial_\mu\rho_\nu\rho^\mu-\partial_\nu\rho_\mu\rho^\nu+\partial_\nu\rho_\mu\rho^\mu \right)$$
We work with lagrangian formalism, so $\rho_\nu$ and $\partial_\mu\rho_\nu$ are independent.
Question 1: Terms 1, 4 and terms 2, 3 of this last expression are not equal, right? We can set the repeated indices to whatever we want, but not the free ones, so $$ \partial_\mu\rho_\nu\rho^\nu\neq\partial_\nu\rho_\mu\rho^\mu $$
Question 2: The derivation of the first term should go like the following:
$$ \frac{\partial}{\partial\rho_\nu}(\partial_\mu\rho_\nu\rho^\nu)= \partial_\mu\rho_\nu\frac{\partial\rho^\nu}{\partial\rho_\nu}=\partial_\mu\rho_\nu\eta^{\nu\kappa}\frac{\partial\rho_\kappa}{\partial\rho_\nu}=\partial_\mu\rho^\kappa\delta_{\kappa\nu}=\partial_\mu\rho^\nu $$
I'm fairly sure this is correct so far. On to the second term, we have
$$ \frac{\partial}{\partial\rho_\nu}(\partial_\mu\rho_\nu\rho^\mu)= \partial_\mu\rho_\nu\frac{\partial\rho^\mu}{\partial\rho_\nu}=\partial_\mu\rho_\nu\eta^{\mu\kappa}\frac{\partial\rho_\kappa}{\partial\rho_\nu}=\partial_\mu\rho_\nu\eta^{\mu\kappa}\delta_{\kappa\nu}=\partial_\mu\rho_\nu\eta^{\mu\nu}=\partial_\mu\rho^\mu $$ This calculation is a bit more involved, and I am unsure whether it's correct. I would like a correction if I missed something.
Provided the two above are correct, the rest should follow in the same fashion, so we don't need to go through the whole calculation.
Thanks in advance!