Prerequisite concepts
Energy and Entropy
The derivative can be interpreted physically
While the constituents of a derivative (the $f$ and $x$ in $\frac{df}{dx}$) can have physical interpretations, the derivative itself can have a different physical interpretation. For example, $\frac{dx}{dt}$ is a velocity. FIXME: find a better example
The negative gradient of the electric potential is the electric field
While the magnitude of the electric field is equal to the gradient of the electric potential, the electric field points in the opposite direction of the gradient of the electric potential, and thus $\vec E = - \vec \nabla V$.