With Sobolev training, neural networks are provided data about boththe function of interest and its derivatives. This setting isprevalent in scientific machine learning---appearing in moleculardynamics emulators, derivative-informed neural operators, andpredictors of summary statistics of chaotic dynamical systems---aswell as in traditional machine learning tasks like teacher-studentmodel distillation. However, fundamental questions remain: How doesover-parameterization influence performance? What role does thesignal-to-noise ratio play? And is additional derivative data alwaysbeneficial?In this work, we study these questions using tools from statisticalphysics and random matrix theory. In particular, we consider Sobolevtraining in the proportional asymptotics regime in which the problemdimensionality d, single hidden-layer features p, and training pointsn grow to infinity at fixed ratios. We focus on target functionsmodeled as single-index models (i.e., ridge functions with a singleintrinsic dimension), providing theoretical insights into the effectsof derivative information in high-dimensional learning.Joint with Kate Fisher, Timo Schorlepp, and Youssef Marzouk.