Formula Derivation Sheet

ROLE: Act as a rigorous mathematics teacher who shows where every formula comes from.

CONTEXT: I teach [CLASS/GRADE] for the [BOARD or EXAM e.g. CBSE / ICSE / State Board / JEE / NEET]. I want my students to UNDERSTAND, not just memorise. I need a derivation sheet for the key formulas of [CHAPTER NAME].

YOUR TASK:
1. Pick the main derivable formulas of the chapter (skip pure definitions).
2. For each, give a clean step-by-step derivation, short but complete, with no skipped algebra.
3. State the starting point or assumption at the top of each derivation.
4. Box or clearly mark the final formula at the end of each derivation.
5. Keep each derivation tight - aim for 4 to 8 lines.

INPUT: Here is the chapter and class. Chapter: [PASTE HERE]. Class/Board: [PASTE HERE].

HOW TO WORK IT OUT: Work each derivation internally and then re-do it line by line to confirm it is valid. VERIFY two ways: (a) check each algebraic step follows from the previous one; (b) confirm the final formula matches the standard textbook formula exactly. Never invent steps, identities, or results. If a derivation is beyond this class level, say so briefly instead of faking it. Keep all maths fully legible in clean plain text or simple LaTeX.

OUTPUT FORMAT (use this exact order):
1. TITLE: '[Chapter] - Derivation Sheet (Class [X], [Board])'.
2. 'Compiled by [YOUR NAME]'.
3. For each formula: a heading, 'Start from:' line, numbered derivation steps, then 'Result:' line with the final formula marked clearly.
4. A short 'Key idea' note under each derivation (one line) on the main trick used.
5. Footer line.

RULES: No skipped algebra. Every step must be valid. Final formula must match the standard form. Keep it to what fits classroom teaching.

OUTPUT FILE FORMAT: present the complete result as a clean, ready-to-use document I can paste straight into Microsoft Word or Google Docs - a clear bold title, section headings, neatly numbered items, comfortable spacing, and simple tables only where helpful (avoid heavy formatting that breaks on paste). It must also print cleanly to PDF on A4 (File > Print > Save as PDF). Keep all mathematics fully readable (fractions as a/b, powers as x^2, roots as sqrt(x)).

MATHS FORMATTING: Write ALL mathematics so it reads correctly in a normal chat and pastes cleanly into Word or Google Docs. Use plain readable notation - for example a/b for fractions, x^2 for powers, sqrt(x) for roots, and symbols like <=, >=, !=, +-, pi, theta, integral and sum. Do NOT output raw LaTeX or code such as \frac{}{}, \sqrt{}, x^{2}, $...$ or \( ... \) - these appear as broken symbols to a teacher in a free chat.

CONTEXT CHECK: Before you begin, if any [SQUARE-BRACKET] above is still empty, or I have not given you the question / photo / data you need, ask me for exactly those and wait - never guess, and never answer with the placeholder text still in brackets.

GROUND RULES (always follow): Do not invent statistics, percentages, marks distributions you were not given, exam cut-offs, solve-time figures, NCERT or textbook page/line numbers, citations, or historical exam data. If you are unsure of any fact or number, say so and clearly label it an estimate; keep established fact separate from your own inference. Every step must be mathematically correct and at the stated class/board/exam level.

SIGNATURE: End your ENTIRE response with exactly this footer block (keep the lines and symbols):
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