Derivatives

• The derivative as instantaneous rate of change and slope of the tangent to y = f(x).
• To compute derivatives from first principles (limit definition) for polynomials.
• The geometric meaning of f′(x) > 0, f′(x) < 0, and f′(x) = 0.
• Physical interpretation: velocity as derivative of position, acceleration as derivative of velocity.

Level 1 — Definition and interpretation

Verbal: The derivative f′(a) measures how fast f(x) changes as x passes through a. On a graph, it is the slope of the tangent at x = a.

Symbolic: f′(x) = lim(h→0) [f(x+h) − f(x)]/h; for xⁿ, f′(x) = n xⁿ⁻¹; (sin x)′ = cos x, (cos x)′ = −sin x.

Symbolic (first principles): f′(x) = lim(h→0) [f(x+h) − f(x)] / h

Example: f(x) = x² f′(x) = lim(h→0) [(x+h)² − x²]/h = lim(h→0) [2xh + h²]/h = lim(h→0) (2x + h) = 2x

Notation: f′(x), dy/dx, Df(x) — all equivalent for y = f(x).

Level 2 — Basic results and applications

Class 11 standard derivatives (from first principles or given):

Physical meaning: If s(t) is displacement (m), v(t) = ds/dt (m/s), a(t) = dv/dt (m/s²).

Increasing/decreasing: f′(x) > 0 → f increasing near x; f′(x) < 0 → decreasing.

Tangent line at (a, f(a)): y − f(a) = f′(a)(x − a).

Differentiability vs continuity: Differentiable at a ⇒ continuous at a. Converse false (e.g., |x| at 0 is continuous but not differentiable).

NCERT spotlight — Differentiability and applications

|x| at 0 is continuous but not differentiable because left and right derivatives differ. For displacement s(t), velocity is ds/dt and acceleration is d squared s/dt squared.

Tangent and normal: Tangent slope is f prime (a). Normal slope is negative reciprocal if f prime (a) is not zero.

Mean value theorem preview: On [a,b], some point c satisfies f prime (c) equals average rate of change — foundation for Class 12 analysis.

Find the derivative of f(x) = 3x² − 5x + 2 from first principles, and the equation of the tangent at x = 1.

Step 1 — f(x+h) = 3(x+h)² − 5(x+h) + 2
         = 3(x² + 2xh + h²) − 5x − 5h + 2.
Step 2 — f(x+h) − f(x) = 3(2xh + h²) − 5h = h(6x + 3h − 5).
Step 3 — [f(x+h) − f(x)]/h = 6x + 3h − 5 → f′(x) = 6x − 5.
Step 4 — At x = 1: f(1) = 0, f′(1) = 1.
Step 5 — Tangent: y − 0 = 1(x − 1) → y = x − 1.

Applications — optimisation sketch

For rectangle perimeter 20, area A = x(10-x) = 10x - x squared. dA/dx = 10 - 2x = 0 gives x = 5, square maximises area. Sign of derivative: positive before x=5, negative after — confirms maximum. Marginal cost in economics is derivative of total cost function with respect to quantity produced.

Deep dive — geometric interpretation and motion problems

Derivative as slope function f prime(x) gives tangent slope at every x — where f prime = 0 horizontal tangent candidate extremum. Normal line slope −1/f prime(a) perpendicular to tangent at (a, f(a)). Increasing/decreasing intervals: f prime > 0 increasing, < 0 decreasing — sign chart from critical points. Particle motion s(t) = t cubed − 6t: v(2) = 12−6 = 6 m/s, a(2) = 6 m/s² — interpret positive velocity direction motion. Differentiability corner |x| at 0 — left derivative −1 right +1; cusp x^(2/3) vertical tangent. Mean value theorem geometric: somewhere tangent parallel secant chord — existence not construction. First principles on x^n establishes power rule for integer n — binomial expansion of (x+h)^n or induction proof in rigorous courses. Board exams may ask 4-mark first principles on simple polynomial — practise full limit steps without skipping h cancellation justification.

Review and practice drill

Review checklist: (1) First principles four steps: f(x+h), subtract, divide, limit. (2) Tangent slope = derivative at point. (3) Positive derivative means increasing locally. (4) Differentiability implies continuity. Practice: Find derivative of f(x) = 1/x at x = 2 using definition — result -1/4.

• Differentiate f(x) = x³ − 4x from first principles.
• A particle has s(t) = t² − 3t. Find velocity at t = 2 s.
• Where is f(x) = x² − 6x + 5 increasing?

Open the Practice tab for graded questions on Derivatives.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.