If you have a Lorentz tensor $T$ with components $T_{\mu\nu}$, it seems clear that $$ \frac{\partial T_{\mu\nu}}{\partial T_{\rho\sigma}} = \delta^\rho_\mu \delta^\sigma_\nu. \tag{1} $$ However, if $T$ is symmetric, so $T_{\mu\nu} = T_{\mu\nu}$, there are two ways this derivative could be nonzero, so $$ \frac{\partial T_{\mu\nu}}{\partial T_{\rho\sigma}} = \delta^\rho_\mu \delta^\sigma_\nu + \delta^\sigma_\mu \delta^\rho_\nu. \tag{2} $$ Does this mean the first expression is wrong? It seems that some information is lost if one naively uses the first expression with a symmetric matrix, because the result lacks the symmetry in $\mu$ and $\nu$ and in $\rho$ and $\sigma$ that the derivative must have.
Additionally, should there be a factor of $\frac{1}{2}$? If one sets $\mu = \nu = \rho = \sigma$, then the second expression yields 2. However, introducing this factor seems to then yield $\frac{1}{2}$ when $\mu = \rho$ and $\nu = \sigma$ but $\mu \neq \sigma$, when it should yield 1. What am I missing?
