First one has to clarify things, because the OP makes no sense to begin with. One should note that in QM ##t## is not an observable and thus not an operator but, of course, the position ##x## is, and it's represented by a self-adjoint operator ##\hat{x}##.
Now in the Schrödinger picture of time evolution (pure) states are represented by time-dependent state vectors ##|\psi(t) \rangle##, which fulfill the Schrödinger equation,
$$\mathrm{i} \hbar \partial_t |\psi(t) \rangle=\hat{H} |\psi(t) \rangle, \qquad(1)$$
where ##\hat{H}## is the Hamilton operator of the system. It's also a self-adjoint operator and represents the total energy of the system. Taking the adjoint of this equation, you get
$$-\mathrm{i} \hbar \partial_t \langle \psi(t) \rangle =\langle \psi(t) |\hat{H}^{\dagger}=\langle \psi(t) | \hat{H}. \qquad (2)$$
Note that in the Schrödinger picture ##\hat{x}## is not dependent on time. So the question is how to define the velocity ##v##, which in classical physics is defined as the time derivative of ##x## along the trajectory of the particle. In QM we like ##\hat{v}## to be a self-adjoint operator for which
$$\langle v \rangle=\frac{\mathrm{d}}{\mathrm{d} t} \langle x \rangle.$$
Now the expectation value for the system in the state represented by ##|\psi(t) \rangle## is given by
$$\langle x \rangle=\langle \psi(t)|\hat{x}|\psi(t) \rangle,$$
and we can calculate the time derivative, using the Eqs. (1) and (2):
$$\frac{\mathrm{d}}{\mathrm{d} t} = (\partial_t \langle \psi(t)|\hat{x}|\psi(t) \rangle + \langle \psi(t) |\hat{x}|(\partial_t \psi(t) \rangle) = \frac{\mathrm{i}}{\hbar} (\langle \psi(t)| \hat{H} \hat{x}|\psi(t) \rangle - \langle \psi(t)|\hat{x} \hat{H}|\psi(t) \rangle=\langle \psi(t)| \mathrm{i}/\hbar [\hat{H},\hat{x}]|\psi(t).$$
Since this should hold true for all ##|\psi(t) \rangle## we conclude that
$$\hat{v}=\frac{\mathrm{i}}{\hbar} [\hat{H},\hat{x}] = \frac{1}{\mathrm{i} \hbar} [\hat{x},\hat{H}],$$
is the operator that represents the velocity. One thus defines a "covariant time dervative" for operators that represent an observable by
$$\mathring{\hat{A}}(\hat{x},\hat{p})=\frac{1}{\mathrm{i} \hbar} [\hat{A}(\hat{x},\hat{p}),\hat{H}].$$
If you have an operator that also explicitly depends on time, then you have of course also the corresponding partial time derivative, i.e.,
$$\mathring{\hat{Q}}(\hat{x},\hat{p},t)
=\frac{1}{\mathrm{i} \hbar} [\hat{Q}(\hat{x},\hat{p}),\hat{H}] + \partial_t \hat{Q}(\hat{x},\hat{p},t).$$