Calculus, as commonly taught, describes certain operations on explicit functions, but science relies on experimental data, which is inherently discrete. In the face of this disparity, how can we help students transition from lower-division mathematics courses to upper-division coursework in other STEM disciplines? We discuss here our efforts to address this issue for upper-division physics majors by introducing a new representation for derivatives in terms of experiments to go along with the traditional symbolic, graphical, verbal, and numerical representations, and by emphasizing infinitesimal reasoning through the use of differentials. These ideas culminate in the concept of thick derivatives. By providing examples of "physics" reasoning about both ordinary and partial derivatives, and methods for incorporating such reasoning into the classroom, we hope to give instructors of calculus new insight into the needs of many of their students.