The marginal likelihood is the gold standard for Bayesian model comparison although it is well-known that the value of marginal likelihood could be sensitive to the choice of prior hyperparameters. 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. In this paper we develop efficient and feasible methods to compute the sensitivities of marginal likelihood, obtained via two common simulation-based methods, with respect to any prior hyperparameter alongside the MCMC estimation algorithm. Our approach builds on Automatic Differentiation (AD), which has only recently been introduced to the more computationally intensive setting of Markov chain Monte Carlo simulation. We illustrate our approach with two empirical applications in the context of widely used multivariate time series models.