Time derivatives of weak values

The time derivative of a physical property often gives rise to another meaningful property. Since weak values provide empirical insights that cannot be derived from expectation values, this paper explores what physical properties can be obtained from the time derivative of weak values. It demonstrates that, in general, the time derivative of a weak value that is invariant under an electromagnetic gauge transformation is neither a weak value nor a gauge-invariant quantity. Left- and right-hand time derivatives of a weak value are defined, and two necessary and sufficient conditions are presented to ensure that they are also gauge-invariant weak values. Finite-difference approximations of the left- and right-hand time derivatives of weak values are also presented, yielding results that match the theoretical expressions when these are gauge invariant. With these definitions, a local Ehrenfest-like theorem can be derived, giving a natural interpretation for the time derivative of weak values. Notably, a single measured weak value of the system's position provides information about two additional unmeasured weak values: the system's local velocity and acceleration, through the first- and second-order time derivatives of the initial weak value, respectively. These findings also offer guidelines for experimentalists to translate the weak value theory into practical laboratory setups, paving the way for innovative quantum technologies. An example illustrates how the electromagnetic field can be determined at specific positions and times from the first- and second-order time derivatives of a weak value of position.