To find potential energy in three dimensions, partial derivatives are used instead of total derivatives because they allow for the analysis of how potential energy changes with respect to each coordinate independently. This approach is essential for deriving the associated vector field, such as gravitational acceleration, from the potential field. In a conservative vector field, one can assign a zero potential at a specific point and use a path integral to determine potential energy across different positions. Total derivatives would require a specific path or tangent, which complicates the analysis in multi-variable contexts. Thus, partial derivatives provide a clearer method for understanding the relationship between potential energy and spatial variables.