From the use of the Minkowski metric, converting between contravariant and covariant
[math]\displaystyle{ \mathbf x_{\mu} \equiv \eta_{\mu}^{\mu} \mathbf x^{\mu} }[/math]
Using matrix multiplication, we have already defined the covariant term,
[math]\displaystyle{ \mathbf{x_{\mu}}= \begin{bmatrix}
dx_0 & -dx_1 & -dx_2 & -dx_3
\end{bmatrix} }[/math]
and the contravariant term
[math]\displaystyle{ \mathbf{x^{\mu}}=
\begin{bmatrix}
dx^0 \\
dx^1 \\
dx^2 \\
dx^3
\end{bmatrix}
}[/math]
Following the rules of matrix multiplication this implies that the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector.
[math]\displaystyle{ \nabla_{\mu}=\partial_{\mu}=\frac{\partial}{\partial x^{\mu}} }[/math]
[math]\displaystyle{ \mathbf \partial_\mu \equiv \Biggl [ \frac{\partial}{\partial t}\quad \frac{\partial}{\partial x}\quad \frac{\partial}{\partial y}\quad \frac{\partial}{\partial z}\Biggr ]=\Biggl [\frac{\partial}{\partial x^0}\quad \frac{\partial}{\partial x^1}\quad \frac{\partial}{\partial x^2}\quad \frac{\partial}{\partial x^3}\Biggr ] }[/math]