Performing calculus: Asymmetric adaptive stimuli-responsive material for derivative control.

Materials (e.g., brick or wood) are generally perceived as unintelligent. Even the highly researched "smart" materials are only capable of extremely primitive analytical functions (e.g., simple logical operations). Here, a material is shown to have the ability to perform (i.e., without a computer), an advanced mathematical operation in calculus: the temporal derivative. It consists of a stimuli-responsive material coated asymmetrically with an adaptive impermeable layer. Its ability to analyze the derivative is shown by experiments, numerical modeling, and theory (i.e., scaling between derivative and response). This class of freestanding stimuli-responsive materials is demonstrated to serve effectively as a derivative controller for controlled delivery and self-regulation. Its fast response realizes the same designed functionality and efficiency as complex industrial derivative controllers widely used in manufacturing. These results illustrate the possibility to associate specifically designed materials directly with higher concepts of mathematics for the development of "intelligent" material-based systems.