# First derivative with multiple indices

Are you really looking for derivatives like that, or is the actual aim to be able to write something like $\delta{S}/\delta{A_{\mu\nu}}$?
Well, I actually need something like $\delta B^{\mu\nu}/{\delta A^{\rho\sigma}}$. At the moment I have to use 'vary' to compute $\delta B^{\mu\nu}$ and then set $\delta A^{\rho\sigma} \to 1$ at the end to extract $\delta B^{\mu\nu}/{\delta A^{\rho\sigma}}$. But then if I have a more complicated expression such as $\delta B^{\mu\nu}/{\delta A^{\rho\sigma}} + B^{\mu\nu}B_{\rho\sigma}$, I cannot simply substitute whatever expression I have for $B^{\mu\nu}$ into it as the derivatives aren't automatically calculated. So I calculate the derivative separately and substitute in the expression. The problem with that is that I still have to define $\delta{\mu\nu} (= \delta/\delta A^{\mu\nu})$ as a derivative so that Cadabra gets the indices right in the substitutions. Since $\delta\{\mu\nu}$ is a second derivative, my calculation would clash if I perform an operation such as product_rule, thus I don't use $\delta_{\mu\nu}$ for anything other than a dummy variable that gets replaced by the derivative I obtained with vary. I was just wondering if there is any easier/cleaner route to this and it seems that a derivative with multiple indices would solve my problem.