# Commutator between covariant derivative and a field

I have field as an element of a Lie algebra as $$\Phi = \phi^at^a$$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$ with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant derivative, where $${t^a}$$ are the generators of the group. When the professor did the calculation explicitly, he did $$[\partial_{\mu}, \Phi] = \partial_{\mu}\Phi - \Phi\partial_{\mu} = (\partial_{\mu}\Phi) + \Phi\partial_{\mu} - \Phi\partial_{\mu}.$$ I don't understand this last step.

Any guidance is much appreciated!

• Remember what the commutator acts on and then use the product rule. Jun 18 at 21:45
• If you like this question you may also enjoy reading this related Phys.SE post. Jun 18 at 22:18

$$[\partial_\mu,\Phi]f=\partial_\mu(\Phi f)-\Phi\partial_\mu f=(\partial_\mu \Phi)f+\Phi \partial_\mu f-\Phi \partial_\mu f.$$
The last step is just applying the product rule for differentiation. The RHS can also be written as $$(\partial_\mu \Phi+\Phi \partial_\mu-\Phi \partial_\mu)f$$, so $$[\partial_\mu,\Phi]=\partial_\mu \Phi+\Phi \partial_\mu-\Phi \partial_\mu$$.