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Now, I need to prove this to myself.

So, here goes nothing.

##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big)##

## = \frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \eta^{\mu\nu}\partial_{\mu} \phi\ \partial_{\nu} \phi \big)##

##= \eta^{\mu\nu}\ \partial_{\nu} \phi + \eta_{\mu\nu} \eta^{\mu\nu}\ \partial_{\mu} \phi##,

where I first differentiated the factor ##\partial_{\mu}\phi## with respect to ##\partial_{\mu}\phi## and then I differentiated the factor ##\partial_{\nu}\phi## with respect to ##\partial_{\mu}\phi##.

Am I correct so far?