This dissertation is in the field of Nonparametric Derivative Estimation using Penalized Splines. It is conducted in two parts. In the first part, we study the L2 convergence rates of estimating derivatives of mean regression functions using penalized splines. In 1982, Stone provided the optimal rates of convergence for estimating derivatives of mean regression functions using nonparametric methods. Using these rates, Zhou et. al. in their 2000 paper showed that the MSE of derivative estimators based on regression splines approach zero at the optimal rate of convergence. Also, in 2019, Xiao showed that, under some general conditions, penalized spline estimators of mean regression functions achieve optimal L2 rates of convergence. We extend this result to derivative estimators. In particular, we show that under similar conditions, penalized spline estimators of derivatives of mean regression functions achieve optimal L2 rates of convergence. In the second part of the thesis, we estimate the amount of association between physical activity and all-cause mortality in US adults using penalized splines. We introduce a novel nonparametric isotemporal substitution model to investigate the dose-response relationship between daily time allocations across physical activity and sedentary behaviors, and all-cause mortality. Our method reveals that the association between such daily time allocations and mortality depends on one’s level of physical activity. We apply our method to data from the 2003-2006 wave of the US National Health and Nutrition Examination Survey (NHANES) with mortality follow-up through December 31st, 2019, a nationally representative survey. Among US adults with less than 6 hours of daily activity, replacing 1 hour of physical activity with sedentary behaviors is associated with up to 68% increase in mortality risks after adjusting for sleep time and baseline demographic and health covariates. In addition, for those with sedentary time above 50% of non-sleep time, replacing 1% of moderate-to-vigorous activity (MVPA) time with sedentary time is associated with up to 18% increase in mortality risks. Therefore, to better understand mortality risk associations, US adults may consider their full daily activity time allocations before replacing one activity type with another.