= Solution
A <Hermitian operator> $A$ satisfies $\langle\varphi,A\psi\rangle=\langle A\varphi,\psi\rangle$ on its <operator domain>. Here
$$
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).
$$
Two <integration by parts> operations, together with the boundary conditions in the Hamiltonian domain, show that the kinetic term is Hermitian; multiplication by the real function $V$ is Hermitian as well. Thus $H$ is Hermitian.
For a time-independent operator $A$, the <Schrödinger equation> and its adjoint give the <Ehrenfest theorem>
$$
\frac d{dt}\langle A\rangle=\frac{i}{\hbar}\langle[H,A]\rangle.
$$
Using the <canonical commutation relation> $[x,p]=i\hbar$,
$$
[H,x]=-\frac{i\hbar}{m}p,
\qquad
[H,p]=i\hbar V'(x).
$$
Consequently
$$
\boxed{\frac d{dt}\langle x\rangle=\frac1m\langle p\rangle,
\qquad
\frac d{dt}\langle p\rangle=-\langle V'(x)\rangle.}
$$
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