Differentiable simulation propagates derivatives through a physics simulator, turning animation, system identification, control, and computational design into gradient-based optimization. Engines differ in the physics they model and, less visibly, in which mathematical object they differentiate. We separate two questions the literature conflates. The referent is which object the returned derivative belongs to: the finite computation that executed, the converged equation it approximates, or the continuous dynamics. The realization describes the organization of the backward pass: automatically generated differentiation, an equation-level adjoint solve, or a custom adjoint. The referent defines the target against which exactness is assessed; the implementation determines how accurately that target is approximated. Cost depends on the realization together with how the states needed by the backward pass are retained or recovered. Crossing the axes places every method and engine we classify in a cell, separates claims the literature merges, and shows that the apparent trade-off between exactness and memory follows from how methods have been built rather than from a constraint. A second, orthogonal taxonomy covers contact, where the map is non-smooth. We then ask whether a correct gradient helps. Four mechanisms by which useful gradients are lost are cataloged and measured on one testbed, where the advantage of the exact derivative over the equation-level adjoint depends on where the fitting target comes from as much as on the solver budget, and we review the evidence that zeroth-order methods can outperform simulator gradients in the regimes where those mechanisms act. The model classes, systems techniques and applications are surveyed through this lens, and we close with a reporting checklist, a released diagnostic that computes it, and a research agenda.

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