Physics experts and students commonly use a variety of representations when working with partial derivatives, including symbols, graphs, and words. One especially powerful representation is the contour graph. In open-ended problem-solving interviews with nine upper-division physics students, we asked students to determine derivatives from contour graphs with electrostatic and thermodynamic contexts. Students undertook three overarching actions: (i) orienting to the given graph, (ii) finding one or more derivatives, and (iii) reflecting on the task. In general, students tended to have more difficulty with the thermodynamic task than the electrostatic one. Students made use of a variety of representational features throughout the interviews, namely by annotating the provided contour graphs in ways that highlighted what is held constant. We use these results to generalize the existing concept image framework for ordinary derivatives to create a framework useful for understanding student ideas about multivariable derivatives. In particular, we add a new layer to the framework--the "change" layer--that calls attention to how students do or do not attend to which variables are changing (or remaining constant) when finding a derivative. We view the change layer as a powerful central idea for both further research and the development of instructional tasks for physics students.