Formula Sheet Link-by-Link Audit

ROLE: Act as a formula-sheet editor who derives, conditions, and independently checks each formula rather than compiling an unexamined list.

CONTEXT: A teacher needs a trustworthy recall sheet whose links help inspect symbolic formulas but do not hide domain, notation, or theorem assumptions.

INPUTS — FILL EVERY PLACEHOLDER:
- [TOPIC OR CHAPTER]
- [EXAM AND LEVEL]
- [REQUIRED FORMULAS]
- [SYLLABUS LIMITS]

DO THIS:
1. Organise formulas by subtopic and define every symbol before first use.
2. State domain, units, sign, branch, and nonzero conditions beside each formula.
3. Give a one-line derivation, definition, or relationship for each formula where feasible.
4. Create a separate approved Symbolab or WolframAlpha check link for every machine-checkable formula.
5. For theorems or geometric claims not settled by symbolic evaluation, provide a human proof cue instead of a fake check.
6. Add one tiny sanity example per formula without replacing the general conditions.
7. Flag convention-dependent formulas explicitly.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Include and check the power-rule formula for differentiating x^n.
Expected result or check target: The derivative is n*x^(n-1), subject to the usual domain and differentiability conditions for the chosen n and number system.
check this yourself
CHECK → https://www.symbolab.com/solver?query=differentiate%20x%5En
(paste-fallback: differentiate x^n)
Encoding note: The space and caret are encoded as %20 and %5E; decoding the Symbolab query gives the fallback exactly.

OUTPUT FORMAT:
1. Notation Key — every symbol, unit, and convention
2. Formula Sheet — grouped formula, conditions, and use case
3. Formula Check Links — one numbered raw URL and exact fallback per formula
4. Mini Sanity Examples — one substitution per formula
5. Human-Proof Cues — formulas not machine-checkable as general claims
6. Common Confusions — sign, branch, units, and look-alike formulas
7. NOT-MACHINE-CHECKABLE — theorem and derivation scope

QUALITY AND TOOL GUARDRAILS:
- Never list a formula without its variable meanings and validity conditions.
- Do not use a few substituted values as proof of a general formula.
- Use plain calculator syntax and the approved tool endpoints only.
- Fully encode each query, label it check this yourself, and include its exact fallback.
- Do not claim official syllabus status beyond the scope the user supplies.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.