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Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$.

Question 1:

Is $\partial_\mu\varphi=(\partial_t+\partial_x)\varphi$ now a vector/tensor, or still a single variable?

Question 2:

Further, is there any difference between $\displaystyle \frac{\partial (\partial_\mu\varphi) }{\partial (\partial_\mu \varphi)}$ (which equals to $1$ I suppose) and $\displaystyle \frac{\partial (\partial^\mu\varphi) }{\partial (\partial_\mu \varphi)}$?

Question 3:

How to take derivative with respect to complex field? i.e. $\displaystyle \frac{\partial (\partial_\mu\varphi^\dagger) }{\partial (\partial_\mu \varphi)}$ , $\displaystyle \frac{\partial (\partial^\mu\varphi^\dagger) }{\partial (\partial_\mu \varphi)}$ and $\displaystyle \frac{\partial \varphi^\dagger}{\partial\varphi}$?

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Correct me if I'm wrong on this, guys, and take this answer with a grain of salt until verified by someone more experienced.

For a scalar field $\phi$ in flat space,

Question 1: The derivative of a scalar field is a vector, which we can see because we pick up an index $\mu$ when we operate $\partial_\mu$ on $\phi$. This is like the gradient of a scalar field in vector calculus, which is a vector.

Question 2: $$\frac{\partial(\partial^\mu\phi)}{\partial(\partial_\mu\phi)}= \frac{\partial}{\partial(\partial_\mu\phi)}(\partial^\mu\phi)=\frac{\partial}{\partial(\partial_\mu\phi)}(g^{\mu\nu}\partial_\nu\phi)=g^{\mu\nu}\frac{\partial(\partial_\nu\phi)}{\partial(\partial_\mu\phi)}=g^{\mu\nu}\delta_\nu^\mu=g^{\mu\mu}$$

Question 3: Treat the conjugate field as independent from the field. That is, the partial derivatives you have provided as examples are all zero.

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