Notation in the Liouville-von Neumann equation $ i\hbar\frac{dI}{dt} = i\hbar\frac{\partial I}{\partial t} + [I, H] $

Question

The Liouville-von Neumannn equation is defined by $$ i\hbar\frac{dI}{dt} = i\hbar\frac{\partial I}{\partial t} + [I, H] $$ where $I$ is any operator and $H$ is the Hamiltonian. I assume that the left-hand side of the equation is a total derivative of $I$ while the right hand side involves a partial derivative of $I$. How is the total derivative of an operator defined in the context of quantum mechanics? Oftentimes, the equation is useful when $\frac{dI}{dt} = 0$; is the relationship derived from the Schrödinger equation?