It is well-known that the marginal likelihood, the gold standard for Bayesian model comparison, can be sensitive to prior hyperparameter choices. However, most models require computationally intense simulation-based methods to evaluate the typically high-dimensional integral of the marginal likelihood expression. Hence, despite the recognition that prior sensitivity analysis is important in this context, it is rarely done in practice. We develop efficient and feasible methods to compute the sensitivities of the marginal likelihood, obtained via two common simulation-based methods, with respect to any prior hyperparameter, alongside the Markov chain Monte Carlo (MCMC) estimation algorithm. Our approach builds on automatic differentiation (AD), which has only recently been introduced to the more computationally intensive MCMC simulation setting. We illustrate our approach with two empirical applications in the context of widely used multivariate time series models.