Deep Gaussian Processes (DGPs) compose Gaussian Process (GP) layers to warp inputs, improving the emulation of computer simulators with nonstationary input-output mappings. While gradient posteriors are analytically available for GPs, quantifying gradient uncertainty for DGPs is more challenging due to their hierarchical composition. In this paper, we propose an efficient approximation to the gradient distribution of a two-layer DGP emulator without requiring simulator gradients. Using the local linearization, we derive closed-form approximations for the gradient mean and covariance, enabling efficient gradient evaluation with uncertainty quantification (UQ). Empirical results show that the proposed approximation provides accurate gradient estimates and informative UQ on nonstationary examples. We then use the derived gradient uncertainty to guide sequential design for computer models with sharp variations. Specifically, we represent sharp-variation regions as super-level sets of the gradient norm and introduce an entropy-based acquisition rule that selects new samples where the estimated sharp-region is most uncertain. Experiments on synthetic benchmarks and a real-world application show that the resulting sequential design improves emulation accuracy for functions with sharp variations relative to existing design methods.

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