derivative-lesson

name	derivative-lesson
description	Turn high-school derivative and calculus application problems into compiled interactive lesson pages in the shuxueshuo repository. Use for tangent or common-tangent problems, monotonicity and extrema analysis, and parameter-range problems solved with derivatives. The agent writes teaching markdown plus declarative calculus JSON; repository tools validate and compile the final HTML.

Derivative Lesson

Create or update a 高中导数及其应用 interactive page. Treat HTML as a compiled artifact: never hand-write final HTML, SVG paths, chart sampling, slider wiring, or page runtime JavaScript for one problem.

Output Contract

Work in this order:

1. Read internal/senior-high/knowledge-points/calculus-methods.md and calculus-case-index.md.
2. Select one primary pattern, the methods actually used by the solution, and up to three similar published cases.
3. Create or update internal/senior-high/lesson-specs/<problem-id>/01_problem.md.
4. Create or update 02_solution.md and 03_visual_steps.md.
5. Create or update:
   • calculus-spec.json
   • calculus-decorations.json
   • lesson-data.json
6. Keep lesson-data.meta.classification synchronized with the final solution.
7. For a publish-ready page, add or update the case in calculus-case-index.md.
8. Validate and compile:

node tools/validate-calculus-spec.mjs internal/senior-high/lesson-specs/<problem-id>/
node tools/build-calculus-page.mjs internal/senior-high/lesson-specs/<problem-id>/

9. Open the compiled HTML directly and inspect every interactive step. Start a local static server only when the browser or an asset-loading requirement makes file:// insufficient.

Required References

• Always read references/calculus-solving-principles.md before writing the solution.
• Always read references/calculus-json-schema-guide.md before writing JSON.
• Always read references/calculus-visualization-principles.md before planning diagrams or interactions.
• Read the real schemas when fields are uncertain; schemas override prose documentation.

Core Rules

• Use the genuine generating variable named by the problem, such as a contact-point abscissa, as the main slider.
• Reuse the problem's original symbol; do not rename an existing x1 to t without a real conflict.
• Do not make the unknown coefficient being solved (a, b, or c) the slider when it is determined by another variable.
• Define auxiliary notation locally. Prefer a familiar name such as h(x1) or phi(x1) and explicitly state a=h(x1); do not present a temporary symbol as though it were universal notation.
• Declare each derivative in functions[].derivativeExpr; the runtime does not perform symbolic differentiation.
• Use ordered bindings for dependent values such as slopes, second contact points, and coefficients.
• Use multiple synchronized panels only when they explain different mathematical spaces, such as the original curves and an auxiliary function a=h(t).
• Remove an earlier panel once later steps no longer use its mathematical information; let the active panel expand instead of carrying decorative context forward.
• Treat graphs as observation and explanation. Complete range and monotonicity proofs with derivative signs, domain boundaries, endpoints, or behavior at infinity.
• Keep one mathematical idea per lesson step. Do not reveal a solved parameter, critical point, or final range before the matching derivation.
• Split an auxiliary-function range proof into four stages: define the function, find critical points, determine monotonicity, then compare global values and conclude the range.
• Use formal full titles on lesson cards and shorter matching labels in step navigation; preserve the same mathematical action in both.
• Every interactive control must change visible mathematical state. Keep representative values clickable even when continuous dragging has no teaching value.
• Keep all JSON declarative and free of HTML fragments.

Supported V1 Scope

• Tangent lines at a point.
• Common tangents and two contact points.
• Monotonicity and local/global extrema.
• Parameter ranges derived from a one-variable auxiliary function.
• Polynomial, exponential, logarithmic, and basic trigonometric expressions supported by the expression engine.

Do not use this skill for symbolic integration, multivariable calculus, complex double-parameter classifications, or a full LLM solver pipeline.

Final Review

• The problem source, title, answers, and output path are correct.
• Function domains are stated before differentiation when needed.
• Every derivative expression is algebraically correct and passes numerical validation.
• Tangent points, slopes, and intercepts remain synchronized while dragging.
• Function names, values, and curves use f, f(x), and y=f(x) consistently; multi-curve panels label every curve directly.
• Every temporary function or point name is defined before use and cannot be mistaken for a standard symbol.
• Slider range is explicitly described as an exploration window when the mathematical variable ranges over all real numbers.
• Critical values and interval endpoints agree across markdown, JSON, chips, and diagrams.
• Auxiliary-function range lessons reveal critical points, derivative signs, and the final range in separate steps.
• A critical-point step shows exactly the named critical positions; it does not add an unrelated free moving point.
• A final comparison step labels every function value used in the comparison, not only the winning extremum.
• Parallel tangent steps use matching derivation structures and ∵ / ∴ connectors; contact-point coordinates appear in the relevant diagram.
• No panel, legend, control, or label remains after it stops supporting the current mathematical action.
• The final range is proved rather than inferred only from the visible chart.
• Step IDs align across lesson data and calculus decorations.
• Validation and compilation succeed without patching generated HTML.