Knowledge of the rate of a biological process is important for characterizing the system and is necessary for gaining a deeper understanding of the process. Consider measurements,Y, made over time on a system following the modelY=f(t) +e, wherefis a smooth, unknown function andeis measurement error. Although most statistical methodology has focused on estimatingf(t) orf′(t), in some applications what is of real biological interest is the relationship betweenfandf′. One example is the study of nitrogen absorption by plant roots through a solution depletion experiment. In this casef(t) is the nitrate concentration of the solution surrounding the roots at timetand –f′(t) is the absorption rate of nitrate by plant roots at timet. One is interested in the rate of nitrate absorption as a function of concentration; that is, one is interested in Φ, where Φ(f) = –f′. Knowledge of Φ is important in quantifying the ability of a particular plant species to absorb nitrogen and in comparing the absorption ability of different crop varieties. A parametric model forfis usually not available, and thus a nonparametric estimate of Φ is particularly appropriate. This article proposes using spline-based curve estimates with the smoothing parameter chosen by cross-validation and suggests a method for obtaining confidence bands using a form of the parametric bootstrap. These methods are used to analyze a series of solution depletion experiments and are also examined by a simulation study designed to mimic the main features of such data. Although the truefis a monotonic function, simulation results indicate that for our specific application, constraining the estimate offto be monotonic does not reduce the average squared error of the rate curve estimate, Φ. Although using a cross-validated estimate of the smoothing parameter tends to inflate the average squared error of the rate estimate, an analysis of a set of solution depletion experiments is still possible. Using the proposed methods, we are able to detect a difference in rate curves obtained under different experimental conditions. This is established by applying an analysis of variance (ANOVA)-like test to the estimated rate curves, where the critical value is determined by a parametric version of the bootstrap, and by examining confidence bands for the difference of two rate cures. This finding is important, because it suggests that the shape of Φ may not be constant under the experimental conditions examined.