Consider a four-vector, with all transformations $\Lambda$ being Lorentz transformations.
Under a Lorentz transformation, the original four-vector goes to its transformed version (tilde) according to:
$\widetilde{x}^\nu=\Lambda^\nu_\rho x^\rho$
According to my textbook, the derivative with respect to the original four-vectors components is given by:
$\frac{\partial }{\partial x^\mu} = [\frac{\partial \widetilde{x_\nu}}{\partial x^\mu}]\frac{\partial}{\partial \widetilde{x^\nu}}$
I don't understand why this is the case. More specifically, I don't understand why there is a lowered index on the first derivative, and then a raised index on the second. Why is that the case? If an explanation without Christoffel symbols or Laplacians could be given, that would be ideal. Basically anything further than the standard metric tensor $\eta_{ab}$ is beyond me.