Does $\partial_\mu =\frac{\partial }{\partial x^\mu}$ or $\partial_\mu =\frac{\partial }{\partial x_\mu}$?

Question

I am looking at the chain rule with covariant and contravariant vectors. I understand why we have: $$df=\frac{\partial f}{\partial x^\mu} dx^\mu$$ (Please correct me if I am wrong) since even though the $\frac{\partial f}{\partial x^\mu} $ seems to have an up index, it transforms like a covariant vector. Now it would make sense to me to then write this as: $$df=\partial_\mu f dx^\mu$$ Meaning that: $$\partial_\mu=\frac{\partial }{\partial x^\mu}$$ However, this feels slightly ambiguous. So is this above notation always (/ever) used and is it common to find the notation: $$\partial_\mu=\frac{\partial }{\partial x_\mu}$$ Which seems like the more naive way to do things? Please explain your answer.

Answer 1

If we set $$\partial_{\mu}= \frac{\partial}{\partial x_{\mu}}$$ Then we would have the weird thing that $ dx^{\mu} \partial_{\mu} $ would not contract correctly - therefore it is a better idea to "define" it the other way.

But as always, what is really the best argument here is to look at the transformation behaviour of something to see whether it is covariant or contravariant. $$ \frac{\partial}{\partial x'^{\mu}}=\frac{\partial x^a}{\partial x'^{\mu}}\frac{\partial}{\partial x^{a}}$$ And something that transforms via $\frac{\partial x^a}{\partial x'^{\mu}}$ should have the Lorentz-index below because it is a covariant vector. Thus it makes sense to write $\partial_{\mu}= \frac{\partial}{\partial x^{\mu}}$.