Differentiation Rules

• The sum, difference, and constant multiple rules for rapid differentiation without first principles every time.
• The product rule and quotient rule for combined functions — JEE staples.
• The chain rule for composite functions f(g(x)).
• To differentiate polynomials, rational functions, and simple trigonometric expressions efficiently.

Level 1 — Linearity and power rule

Verbal: Differentiation is linear: the derivative of a sum is the sum of derivatives, and constants factor out.

Symbolic:

• (u ± v)′ = u′ ± v′
• (cu)′ = c u′
• (xⁿ)′ = n xⁿ⁻¹ (n real, NCERT focus on integer n)

Examples:

• d/dx(4x³ − 7x + 1) = 12x² − 7
• d/dx(√x) = d/dx(x^(1/2)) = (1/2)x^(−1/2) = 1/(2√x)

Level 2 — Product, quotient, and chain rules

Trig derivatives (with chain):

• (sin x)′ = cos x; (cos x)′ = −sin x
• (tan x)′ = sec²x; (cot x)′ = −cosec²x

Strategy: Identify outer and inner functions for chain rule. For rational functions, quotient rule or simplify first.

NCERT Class 11 scope: Emphasis on product and quotient for low-degree polynomials with sin/cos; full chain rule extended in Class 12.

NCERT spotlight — Chain rule and beyond

For y = sin(3x), inner function 3x gives dy/dx = 3 cos(3x). Quotient rule on (sin x)/x produces (x cos x - sin x)/x squared. Parametric motion uses dy/dx = (dy/dt)/(dx/dt).

Implicit differentiation preview: From x squared + y squared = 25, obtain dy/dx = -x/y without solving for y explicitly.

Critical points: Solve f prime (x) = 0 to locate candidates for maxima and minima in optimisation problems.

Differentiate y = (2x + 1)³ · sin x.

Step 1 — Product rule: y′ = u′v + uv′ with u = (2x+1)³, v = sin x.
Step 2 — Chain on u: u′ = 3(2x+1)² · 2 = 6(2x+1)².
Step 3 — v′ = cos x.
Step 4 — y′ = 6(2x+1)² sin x + (2x+1)³ cos x
         = (2x+1)² [6 sin x + (2x+1) cos x].
Step 5 — Factorisation helps checking at x = 0: y′(0) = 1²[0 + 1·1] = 1.

Applications — related rates

Balloon radius r increasing at dr/dt = 2 cm/s. Volume V = (4/3) pi r cubed gives dV/dt = 4 pi r squared dr/dt. At r = 10 cm, dV/dt = 800 pi cm cubed/s. Chain rule links time rates of linked variables — classic JEE Main problem template requiring implicit differentiation of geometric formulas.

Deep dive — combined rules and optimisation setup

Differentiate y = x squared e to the x: product rule gives 2x e^x + x squared e^x = x e^x (x + 2). Quotient on tan x = sin x / cos x recovers sec squared x via quotient rule — verify against known derivative. Chain rule nested: sin(ln x) → derivative cos(ln x) × (1/x). Logarithmic differentiation for y = x to the x: ln y = x ln x → (1/y) y prime = ln x + 1 → y prime = x^x (ln x + 1). Second derivative f double prime for concavity: f double prime > 0 concave up. Velocity acceleration: if s = t cubed − 6t, v = 3t squared − 6, a = 6t — jerk is third derivative preview. Tangent normal problems combine rules with point evaluation — always find f prime then substitute x0. Class 12 extends to inverse trig, exponential log, implicit — Class 11 rules are foundation drilling product quotient chain until automatic.

Review and practice drill

Review checklist: (1) Product rule: first d second plus second d first. (2) Quotient rule: low d high minus high d low over low squared. (3) Chain rule: outer derivative times inner derivative. (4) Differentiate polynomials termwise. Practice: y = (x^2 + 1)^3 — chain gives 3(x^2+1)^2 times 2x.

• Find dy/dx if y = x²/(x + 1).
• Differentiate f(x) = cos(5x − 2).
• If y = x·e^x (preview), product rule gives y′ = e^x + x e^x — verify pattern with (uv)′.

Open the Practice tab for graded questions on Rules.

• 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.