One must be careful with the time dependence of operators in quantum mechanics. The mathematical time dependence is arbitrary in a wide range since you can always do time-dependent unitary transformations on the operators that represent observables and states (either represented by normalized Hilbert-space vectors or statistical operators; here I'll use state vectors, but the same arguments also apply to the general case of stat. ops.):
\hat{A}' = \hat{U}(t) \hat{A} \hat{U}^{\dagger}(t), \quad |\psi' \rangle =\hat{U}(t) |\psi \rangle.
So you can shuffle the time dependence from the states to the operators and vice versa without changing the physical results.
In non-relativistic quantum theory of one particle ("1st quantization") with spin 0 (scalar particle) you have the basic observables \vec{x} and \vec{p}. Any other observable is a function of these.
The dynamics of the system is characterized by a Hamiltonian, H(\vec{x},\vec{p}) (where I assumed a Hamiltonian that is not explicitly time dependent, i.e., which only depends on time through its dependence on the fundamental operators \vec{x} and \vec{p}). As stressed above, the mathematical time dependence of the operators and states is arbitrary to a large degree, and you can choose any "picture of time dependence" you like to solve a problem.
Usually one starts in the Schrödinger picture, where the observables are time-independent (except when you consider explicitly time dependend observables, but this we won't do in this posting for sake of clarity). The entire time dependence is thus put to the state vectors. This means you have
\frac{\mathrm{d} \vec{x}}{\mathrm{d} t}=0, \quad \frac{\mathrm{d} \vec{p}}{\mathrm{d} t}=0, \quad \frac{\mathrm{d}}{\mathrm{d} t} |\psi(t) \rangle = -\frac{\mathrm{i}}{\hbar} H |\psi(t) \rangle.
Independently from the choice of the picture of time evolution you may ask, which operator describes the time derivative of an observable. E.g., you may ask, which operator represents the velocity, which in classical physics is the time derivative of the position vector. In the Schrödinger picture, the mathematical time derivative of the position-vector operator is however 0, and thus this time derivative cannot represent velocity. The correct answer is given by the analogy with analytical mechanics (Hamilton formalism), where the time derivative of an observable is given by the Poisson bracket of the observable with the Hamiltonian. The Poisson brackets of classical canonical mechanics map (up to an imaginary factor) to commutators in quantum theory. Thus the correct "physical time derivative" of the position vector, giving the operator, representing velocity, is
\vec{v}=\mathrm{D}_t=\frac{1}{\mathrm{i} \hbar} [\vec{x},H].
This holds true in any picture of the time evolution.
Another picture is the Heisenberg picture, where the state vectors are constant and the entire mathematical time dependence is put to the operators, representing observables. The formal solution of the state-vector equation of motion in the Schrödinger picture is
|\psi(t) \rangle_S=\exp \left (-\frac{\mathrm{i}}{\hbar} t H \right ) |\psi(0) \rangle_{S} = U_S(t) |\psi(0) \rangle_S.
The operator U_S(t) obviously is unitary, because H is self-adfjoint. Thus we can do a transformation to the Heisenberg picture (assuming that state vectors and observable operators coincide in both pictures at the initial time t=0) by setting
|\psi(t) \rangle_S=U_s(t) |\psi \rangle_{H},
where |\psi \rangle_H=|\psi(t=0) \rangle_S is the time-independent state vector in the Heisenberg picture.
The observable operators in the Heisenberg picture become time dependent, because we have to set
A_S=U_S(t) A_H(t) U_S^{\dagger} \; \Leftrightarrow\; A_H(t)=U_S^{\dagger}(t) A_S U_S(t).
Now we can take the mathematical time derivative of this expression. For a observable A that is not explicitly time dependent we get
\frac{\mathrm{d}}{\mathrm{d} t} A_{H}(t)=\frac{\mathrm{i}}{\hbar} U_S^{\dagger} (H A_S-A_S H) U_S=\frac{\mathrm{i}}{\hbar} [H,A_{H}(t)].
Here we have used that
H_S=H_H=H,
because U_S(t) commutes with H. From the previous equation we see that in the Heisenberg picture (and only in the Heisenberg picture) the physical time derivative of observable operators coincides with the mathematical time derivative.
Your original question, is not so much related to this sometimes a bit confusing issue with the time dependence of quantities in quantum theory but with the physical meaning of the momentum operator. That the momentum operator is given as the gradient of the wave function in position representation comes from the fact that momentum is the generator of spatial translations.
