How to take derivative with respect to Lagrangian of complex field?

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 $$\displaystyle \frac{\partial \varphi^\dagger}{\partial\varphi}$$?

• – Qmechanic Oct 14 at 17:53

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