The 4D Gradient Operator ${\partial\over \partial x_\mu}$

This also transforms like a vector for us. For example the derivative of a scalar function $f$, ${\partial f\over\partial x_\mu}$ is a 4-vector. We do need to be careful about the sign on the time derivative. In GR, we keep track of the signs using the metric tensor $g_{\mu\nu}$ but we do not wish to introduce this complication here.

We choose to use the $ict$ crutch to keep the calculations simple for now. If $x_0=ict$, one can see that ${\partial\ \over\partial x_0}$ introduces a sign change due to the $i$ in the denominator. (In GR... vectors can have upper or lower indices and we use the metric tensor to raise and lower them.)