CVP verified calculation capability

Numerical derivative Calculation Capability

Forward, backward, or central finite difference of a polynomial, sin, or exp, plus a numeric two-sided limit. Discovery /two-sided-limit. Not symbolic d/dx.

Math · Numerical Capability ID math.numerical.derivative version 1.1.0 CVP VERIFIED CURRENT Core

Model

Forward

Forward: (f(x+h) − f(x)) / h
Backward: (f(x) − f(x−h)) / h
Central: (f(x+h) − f(x−h)) / (2h)
Two-sided limit: [f(x+h) + f(x−h)] / 2

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`method`	—	—	input / output
`integrand`	—	—	input / output
`x`	—	—	input / output
`h`	—	—	input / output
`coeffs`	c0,c1,…,cn constant first	—	input
`value`	—	—	output
`abs_err`	—	—	output
`quantity`	—	—	output
`left`	—	—	output
`right`	—	—	output
`gap`	—	—	output

Trusted Math contract

Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.

Field	Declaration
`numeric_domain`	real
`exactness`	float64
`algorithm`	forward / backward / central finite difference; two-sided numeric limit
`valid_domain`	finite x,h; published integrand / expression family
`singularity_policy`	INVALID_INPUT when h = 0; limit may be coded as absent
`branch_policy`	not applicable — real iterate
`tolerance`	note: step h is an input; truncation vs roundoff is not auto-balanced
`convergence`	not applicable — one-shot stencil, not an adaptive tableau
`conditioning`	too-small h is roundoff-dominated; too-large h is truncation-dominated
`rounding`	ieee754_round_ties_to_even
`overflow`	ieee754_nonfinite_or_coded_error
`reproducibility`	deterministic algorithm + Capability@Version
`independent_oracle`	cvp:math.numerical.derivative O2 / Evidence published cases
`golden_vectors`	Evidence golden + boundary + property (and published_reference_case when listed)

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

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

Example

One numerical check. The calculator page holds worked steps and FAQ.

central poly 0,0,1 at x=3 → 6

Open this example in the calculator

Trust

CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Verified scope: all published methods
Assurance: Core
Declared partition coverage: PASS · 5/5 declared partitions (central, forward, backward, limit, invalid-domain)
Calculation version: 1.1.0
Verification: 14/14 calculation tests · 4/4 legacy golden · 3/3 legacy boundary · 7/7 legacy property
Production surface contract: 4/4
Versions: Calculation 1.1.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.numerical-cvp
Verification revision: 2026-09-22.numerical-cvp · 1/1 property · digest 342f8f495963
Reference: O1 model · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
CVP suite: 5/5 golden · 2/2 CVP boundary · 5/5 invalid · 1/1 property · 3/3 cross-interface · 1/1 cross-calculator · 4/4 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.1.0 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.1.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 4/4.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/numerical-derivative' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"method":"central","integrand":"poly","coeffs":[0,0,1],"x":3},"calculation_version":"1.1.0"}'

{
  "inputs": {
    "method": "central",
    "integrand": "poly",
    "coeffs": [
      0,
      0,
      1
    ],
    "x": 3
  },
  "calculation_version": "1.1.0"
}

POST /api/v1/calc/numerical-derivative. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: numerical-derivative
Engine: numerical-derivative
REST: POST /api/v1/calc/numerical-derivative
MCP: execute_calculation with capability_id: "math.numerical.derivative"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.1.0)
Versioned output schema: Immutable pin (1.1.0)

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-2:2019 Quantities and units — Mathematics
CRC Standard Mathematical Tables — forward, backward, and central differences

Reference evidence classes: O1 model · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.numerical.derivative"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"math.numerical.derivative","inputs":{"method":"central","integrand":"poly","coeffs":[0,0,1],"x":3},"calculation_version":"1.1.0"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.1.0) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.