[QUE/QM-09004]

Without assuming that the Hamiltonian is independent of time, start from the requirement, \[ \begin{equation}
  _H{\langle \psi t \vert X_H(t)}_H\vert \psi \rangle_H = _{S\hspace{-1pt}}\langle\vert {\psi t}{X_S}\vert \psi\rangle_S \end{equation}\]  and directly derive the equation of motion \[ \begin{equation} \boxed{\frac{d X_H}{d t} = \frac{\partial X_H }{\partial t}+  \frac{1}{i\hbar}\big[X_H, H\big]_-.} \end{equation}\] for the general case when the Hamiltonian depends on time.