Time series associated with single-molecule experiments and/or simulations contain a wealth of multiscale information about complex biochemical systems. However efficiently extracting and representing useful physical information from these time series measurements can be challenging. We demonstrate how Penalized splines (P-Splines) can be useful in summarizing complex single-molecule time series data using quantities estimated from the observed data. A design matrix that simultaneously uses noisy function and derivative scatterplot information to refine function estimates using P-spline techniques is introduced. The approach is called the PuDI (P-Splines using Derivative Information) method. We show how Generalized Least Squares fits seamlessly into the PuDI method; several applications demonstrating how inclusion of uncertainty information improves the PuDI function estimates are presented. The PuDI design matrix can be used to assist scatterplot smoothing applications where both unbiased function and derivative estimates are available.