Accounting for the uncertainty in the predictions of modern neural networks is a challenging and important task in many domains. Existing algorithms for uncertainty estimation require modifying the model architecture and training procedure (e.g., Bayesian neural networks) or dramatically increase the computational cost of predictions such as approaches based on ensembling. This work proposes a new algorithm that can be applied to a given trained neural network and produces approximate prediction intervals. The method is based on the classical delta method in statistics but achieves computational efficiency by using matrix sketching to approximate the Jacobian matrix. The resulting algorithm is competitive with state-of-the-art approaches for constructing predictive intervals on various regression datasets from the UCI repository.