Derivative Notation Question (Contravariant vs. Covariant)

Question

I know how to write a covariant derivative in Leibniz notation: $$\partial^\mu\equiv \frac{\partial}{\partial x^\mu}$$ Does that mean that a contravariant derivative in Leibniz notation would be $$\partial_\mu\equiv \frac{\partial}{\partial x_\mu}\tag{Does this expression even make sense?}$$ Another guess: $\partial_\mu\equiv dx_\mu$. The idea is to not have to write something like this: $$g_{\mu\nu}\frac{\partial}{\partial x^\mu}$$ Is there a shorthand that does not write out the metric tensor explicitly, i.e. with the raising and lowering of indices? or does this not exist, since no one writes einstein-notation partials in Leibniz form?

Answer 1

According to my Electrodynamics book, $$\partial _{\mu} = g_{\mu \nu}\partial ^{\nu}$$ where $g_{\mu \nu}$ is metric tensor.