In the context of Higgs physics when solving the Euler-Lagrange equation I end up with derivatives like the following
$\frac{\partial A^\nu}{\partial A^\mu}$
$\frac{\partial A_\mu}{\partial A^\mu}$
Could someone explain to me how to resolve these?
--- Update -----
Further the following derivate occurs
$\frac{\partial (\partial^\mu A^\nu)}{\partial (\partial^\nu A^\mu)}$
This one is obviously much harder than the previous ones. While I feel like getting the clue for the previous ones, I have a hard time resolving this one. As far as I know the result should be a scalar, looking at the Indices.
---- Update 2 -----
For anyone interested, the answers for the first two can be found below. For the derivate introduced in the first update, I decided to change the indices in the Euler Lagrange equation leading to the derivate
$\frac{\partial (\partial^\mu A^\nu)}{\partial (\partial^{\nu^, } A^{\mu^,})}$
This can then be easily found to be
$\delta_{\nu^,}^\mu \delta_{\mu^,}^\nu$
Defined to give kronecker deltas.
The first is $\delta_\mu^\nu$ and the second contracts the indices to form the trace of the identity, yielding D, the space time dimension.