{
  "capability_id": "math.numerical.derivative",
  "tool_id": "numerical-derivative",
  "engine": "numerical-derivative",
  "calculation_version": "1.1.2",
  "generated_at": "2026-10-17T06:06:06.037Z",
  "engine_build": "2026-10-14",
  "test_count": 14,
  "passed": 14,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "passed": 4,
      "total": 4
    },
    "boundary": {
      "passed": 3,
      "total": 3
    },
    "property": {
      "passed": 7,
      "total": 7
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/math/numerical-derivative",
  "api": "https://www.calculatorx.com/api/v1/calc/numerical-derivative",
  "schema_input": "https://www.calculatorx.com/schemas/math.numerical.derivative.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.numerical.derivative.output.json",
  "assumptions": [
    "The Result Card includes an engine-linked figure of f around x (tangent slope is the published finite difference; two-sided limit draws a horizontal at the probe). Same fnValue as REST/MCP. Not symbolic CAS d/dx.",
    "Not one-sided limits",
    "Not L’Hôpital",
    "Polynomial degree ≤ 8"
  ],
  "limitations": [
    "Not symbolic / CAS d/dx",
    "Not one-sided left/right; not L’Hôpital",
    "h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "Polynomial degree ≤ 8",
    "Not a Statistics calculator"
  ],
  "boundary_conditions": [
    "exp(x) underflow to zero → NUMERICAL_RANGE_ERROR",
    "Non-finite function probes, derivative or error → NUMERICAL_RANGE_ERROR",
    "h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "h > 1 → VALUE_OUT_OF_RANGE",
    "unknown method (including autodiff, left, lhopital, cas) → INVALID_MODE"
  ],
  "sources": [
    {
      "title": "ISO 80000-2:2019 Quantities and units — Mathematics",
      "url": "https://www.iso.org/standard/64973.html",
      "supports": "Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.",
      "kind": "standard",
      "publisher": "International Organization for Standardization",
      "reference_scope": "all"
    },
    {
      "title": "CRC Standard Mathematical Tables — forward, backward, and central differences",
      "supports": "Forward (f(x+h)−f(x))/h; backward; central (f(x+h)−f(x−h))/(2h); two-sided probe of f",
      "kind": "reference",
      "reference_scope": "model",
      "oracle_class": "O1"
    },
    {
      "title": "NIST DLMF §3.4 — Differentiation",
      "section": "§3.4",
      "url": "https://dlmf.nist.gov/3.4",
      "supports": "Numerical differentiation using differences, including truncation and rounding considerations.",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
    },
    {
      "title": "Gilbert Strang, Calculus — Chapter 2: Derivatives (MIT OpenCourseWare)",
      "section": "§§2.6–2.7 Limits and Continuous Functions",
      "url": "https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_ch02.pdf",
      "supports": "Two-sided and one-sided limits, continuity, and functions whose limits do not exist. Finite numerical samples alone are not proof of a mathematical limit; CalculatorX’s sampling limitations are stated separately. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "Wellesley-Cambridge Press; hosted by MIT OpenCourseWare"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:dd73311df2b6e31cc00bf131927ce9ba20344f1f597a61c360d3739de4f219e2",
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 14,
      "total": 14,
      "golden": 4,
      "boundary": 3,
      "property": 7
    },
    "contract_checks": {
      "passed": 4,
      "total": 4
    }
  },
  "evidence_core_generated_at": "2026-10-14T05:05:51.930Z",
  "rendered_at": "2026-10-17T06:06:06.037Z",
  "release_integrity": {
    "status": "verified",
    "scope": "workspace",
    "production_attestation_required": true,
    "calculation_version": "1.1.2",
    "artifacts": {
      "engine": "1.1.2",
      "page": "1.1.2",
      "capability": "1.1.2",
      "evidence": "1.1.2",
      "input_schema": "1.1.2",
      "output_schema": "1.1.2",
      "mcp_contract": "1.1.2"
    },
    "digests": {
      "input_schema_sha256": "26040aaa592d3154ca216c017b3ec03ca1e3340e9a03c0e8e8e6ac2593b60a05",
      "output_schema_sha256": "d9d7d84992ee8d9667cc38dfba13bfc7d4f32be89a5f53deb878ba4957b28b4b",
      "evidence_sha256": "4db3bec4b317cf428ed35aaef741cc8817cc2121ad493f2953459716bf1e99e3",
      "evidence_core_sha256": "4db3bec4b317cf428ed35aaef741cc8817cc2121ad493f2953459716bf1e99e3",
      "capability_sha256": "e7735c337f84119aa8a71321faa831c7dc0e8c69b04459a3c53a737b1ac679a7",
      "engine_sha256": "6ef38e64f6a7018cce0538e78ba57c8e2a141f11e8779e9e3f91408a668b7b71",
      "mcp_contract_sha256": "319b7d8e97c240413a4f161888c3ae885a1fa211730c6887ce4f6dfe18a43e37"
    },
    "contract_checks": {
      "total": 4,
      "passed": 4,
      "failed": 0,
      "items": [
        {
          "id": "release-artifacts-aligned",
          "status": "pass",
          "detail": "1.1.2"
        },
        {
          "id": "schema-output-required-declared",
          "status": "pass"
        },
        {
          "id": "schema-rejects-incomplete-input",
          "status": "pass"
        },
        {
          "id": "rest-success-payload-validates-schema",
          "status": "pass"
        }
      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  },
  "tests": [],
  "disclosure": {
    "policy": "calculatorx-public-evidence-v1",
    "full_validation": "private",
    "representation": "summary"
  }
}