Which indices go where in the resulting delta function

Question

I want to understand what the following is equal to in terms of delta functions: $\dfrac{\partial(\partial_\alpha A_{\beta})}{\partial(\partial_\mu A_\nu)}$

I know this is equivalent to: $\dfrac{\partial\biggl(\dfrac{\partial A_\beta}{\partial x^\alpha}\biggr)} {\partial\biggl(\dfrac{\partial A_\mu}{\partial x^\nu}\biggr)}$

And my question is whether it is equal to $\delta^{\mu}_\beta \delta^{\alpha}_\nu$, or $\delta^{\mu}_\alpha \delta^{\beta}_\nu$, or something else, and why.

Thanks a lot for your time

Answer 1

Note that $\partial_\alpha A_\beta=\delta_\alpha^\mu \delta_\beta^\nu\partial_\mu A_\nu$, so the derivative is $\delta_\alpha^\mu \delta_\beta^\nu$, your second suggestion. It could never have been the first one, since $\delta^\alpha_\nu \delta^\beta_\mu\partial_\mu A_\nu$ doesn't have the $\mu$ or $\nu$ indices in the right positions to contract.