We developed the Partial Derivative Machine (PDM) in response to difficulties we encountered in teaching students about mathematical concepts involving partial derivatives and total differentials that are needed in thermodynamics. The Partial Derivative Machine is a system that has four observable and controllable properties: two forces and two positions. However, of these four properties only two may be controlled independently. This context-dependence of independent and dependent variables enables the same sort of mathematical flexibility (and confusion) that is present in thermodynamics. Because the PDM is easy to use and understand, we hypothesized that it would allow us to explore the nature of experts' thinking about derivatives, even those unfamiliar with ideas in thermodynamics. In this paper, we present results from interviews with experts from several disciplines, as we explore how they understand partial derivatives when given an ambiguous prompt. The research question guiding this work is "How do experts think about partial derivatives?"