Survival models --- particularly those able to account for patient comorbidities via competing risks analysis --- offer valuable prognostic information to clinicians making critical decisions and represent a growing area of application for machine learning approaches. However, current methods typically involve restrictive parameterisations, discretisation of time or the modelling of only one event cause. In this paper, we highlight how general cumulative distribution functions can be naturally expressed via neural network-based ordinary differential equations and how this observation can be utilised in survival analysis. In particular, we present DeSurv, a neural derivative-based approach capable of avoiding aforementioned restrictions and flexibly modelling competing-risk survival data in continuous time. We apply DeSurv to both single-risk and competing-risk synthetic and real-world datasets and obtain results which compare favourably with current state-of-the-art models.