A JVP propagates one input direction through the program

For a differentiable map f from R^n to R^m, the input x has n components and its tangent seed v has the same input shape. The output f(x) and tangent J_f(x)v have m components. The Jacobian need never be materialized to compute this product. At each primitive, a derivative rule maps the incoming tangents to outgoing tangents while using the required primal values. Tangent propagation is linear in v at fixed x, even when f is nonlinear in x. This distinction is central to composing correct derivative transformations.

For a product z=ab, the tangent is b times the tangent of a plus a times the tangent of b. For a composition, the local tangent rules compose in program order. These rules differentiate the represented computation within their valid real domains; they do not remove floating-point rounding or make an invalid logarithm differentiable. Separate automatic differentiation from symbolic expression simplification and finite-difference perturbation. AD uses local rules at the represented values, while numerical checks later compare that result against independent evidence.

A single JVP answers a directional question. Recovering every Jacobian column with forward mode requires basis seeds or an equivalent batched transformation, with cost depending on the program and vectorization strategy. One directional propagation commonly costs a small multiple of evaluating the program, but this is not a universal constant-time promise for every primitive, batch layout, or runtime. Forward mode is especially natural when only a few input directions are needed; the best choice depends on the derivative product actually requested and the implementation's memory and operation costs.

The seed is held fixed when defining the directional derivative at x. If a later higher-order calculation makes v depend on x, differentiating J_f(x)v(x) also differentiates that seed unless the program explicitly treats it as constant. Array shapes and broadcasting can conceal this difference: a scalar seed expanded across several inputs asks for a sum of directional contributions, not an identity matrix. JAX's official jax-v0.8.0 autodiff documentation describes JVPs as a first-class transformation; runtime details should be checked against that pinned version when translating the mathematics into code.

Write down one small primal trace, its tangent trace, and the resulting dimensions. Test a zero seed and a scaled seed to catch accidental nonlinear seed handling. Compare independent directional finite differences at smooth, valid points, keeping perturbation scale and rounding in mind. A correct local rule can still be composed incorrectly if a wrapper detaches a value, casts to an unsupported type, or changes an input shape. Inspect the actual transformed path and state the real-valued domain before interpreting a plausible-looking tangent as proof.

The function lab supplies an analytic scalar example, not an AD engine. Reset, set BOTH coefficient a=1.5 and position x=0.5, predict slope 2ax=1.5, then Play or Step. Reset and re-set BOTH a=1.5 and x=1; predict slope three. A tangent seed of −2 would give scalar JVPs −3 and −6 on paper. Edits retain elapsed time; playback traces the fixed quadratic and does not change the selected input or run tangent code. Use its displayed slope to inspect this one rule, not to claim the lab differentiates arbitrary programs.