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I am teaching myself QFT, and this is not a homework problem. Right now, I am learning Classical Field Theory, and I am not sure how to proceed with the following problem. I would be grateful to receive inputs and suggestions.

**Problem**Given the Klein Gordon Lagrangian with a complex field,

[tex]\mathcal{L} = \partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - m^2 \Phi^{\dagger}\Phi[/tex]

find the conserved current and charge.

My problem is two fold:

1. How do I differentiate the first term with respect to [itex]\partial_{\mu}\Phi[/itex] (the term with the conjugate)? I know this is probably a trivial question, but I am new to this notation.

2. What is the significance of a complex field, as opposed to a real field?

Thanks.