A current research problem in mathematics education is the characterization of the role of teachers in the processes of technology integration in mathematics classrooms. This article shows how two secondary mathematics teachers taught the concept of derivative of a function at a point and the concept of derivative function, one of them using digital technology and the other one without using it. Their teaching was characterized by describing their hypothetical learning trajectories (learning goals, learning activities and the hypothetical learning processes). APOS theory (which stands for "Action, Process, Object and Schema") was used to describe the hypothetical learning processes. The results showed that the use of digital technology in class may promote reflection among students without excessive computations, thus helping them to construct the concept of derivative.