Lesson Overview
Subject
Mathematics
Chapter
8 — Differentiation
Reference
S. Chand ISC XII · Art. 8.1–8.27
Schedule
May–Jun 2026 · 7 Days · 49 Questions
How to Use Each Question
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Key Formulae Summary
Core Derivative Rules
First Principles\(\displaystyle f'(x)=\lim_{h o 0}\dfrac{f(x+h)-f(x)}{h}\)
Power\(\dfrac{d}{dx}(x^n)=nx^{n-1}\)
Product\(\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\)
Quotient\(\dfrac{d}{dx}\!\left(\dfrac{u}{v}
ight)=\dfrac{v\,u'-u\,v'}{v^2}\)
Chain\(\dfrac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)
log\(\dfrac{d}{dx}[\log f(x)]=\dfrac{f'(x)}{f(x)}\)
e^x\(\dfrac{d}{dx}(e^{f(x)})=e^{f(x)}\cdot f'(x)\) ; \(\dfrac{d}{dx}(a^x)=a^x\log_e a\)
sin⁻¹\(\dfrac{d}{dx}(\sin^{-1}x)=\dfrac{1}{\sqrt{1-x^2}}\) ; \(\dfrac{d}{dx}(\cos^{-1}x)=\dfrac{-1}{\sqrt{1-x^2}}\)
tan⁻¹\(\dfrac{d}{dx}( an^{-1}x)=\dfrac{1}{1+x^2}\) ; \(\dfrac{d}{dx}(\cot^{-1}x)=\dfrac{-1}{1+x^2}\)
Parametric\(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\)
2nd Order\(\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\!\left(\dfrac{dy}{dx}
ight)\)
Chapter Road Map
DAY 1 · Q1–6
First Principles · Power · Product · Quotient Rules
DAY 2 · Q7–13
Chain Rule · Logarithmic Functions (4 types)
DAY 3 · Q14–19
Exponential Functions · ITF Basics
DAY 4 · Q20–29
ITF Trig Substitution (10 worked types)
DAY 5 · Q30–35
Implicit Differentiation incl. ∞-series
DAY 6 · Q36–41
Parametric · Logarithmic Differentiation
DAY 7 · Q42–49
Second Order Derivatives · ISC Proofs
TOTAL
49 Questions · Art. 8.1–8.27 · Full ISC Coverage
⚠ Common Misconceptions
✗Forgetting Chain Rule when differentiating composite functions — always peel the outer layer and multiply by inner derivative
✗Implicit Differentiation: every \(y\)-term must carry \(\dfrac{dy}{dx}\) as a factor (Chain Rule)
✗ITF substitution: always verify the domain before simplifying \(\sin^{-1}(\sin heta)= heta\)
✗Log Differentiation: take \(\log y\) first, THEN differentiate — do NOT differentiate before taking log
✗Parametric: \(dx/dt
eq 0\) must hold; always express \(dy/dx\) fully in terms of the parameter
DAY 1 · MON
Day 1 — First Principles & Standard Rules
Sections 1–2 — First Principles & Standard Rules
First Principlesf'(x) = lim[h→0] [f(x+h)−f(x)]/h
Powerd/dx(xⁿ) = nxⁿ⁻¹
Product(uv)' = uv' + vu'
Quotient(u/v)' = (vu'−uv')/v²
✎Concept Board — Day 1Click to open · Draw diagrams & explain concepts▶
Q1
\(\dfrac{x+1}{2x-3}\) from first principles (4M)
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Q2
\(x^{3/4}\) using Power Rule (1M)
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Q3
\(x^2\cos x\) — Product Rule (2M)
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Q4
\(x\sin x\log x\) — Product Rule for three factors (3M)
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Q5
\(\dfrac{x^2+1}{x^2-1}\) — Quotient Rule (3M)
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Q6
\(\dfrac{x^3-x}{x}\) — Simplify then differentiate (2M)
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📎 Assignments — Day 1 (Optional · Viewable by students)
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Day 1 — Video Resources
Exit Ticket
1State the first-principles definition of f'(x).
2Use Power Rule to find dy/dx when y=x³.
Key Formulas — Day 1
•(xⁿ)' = nxⁿ⁻¹
•Product: (uv)'=uv'+vu'
•Quotient: (u/v)'=(vu'−uv')/v²
DAY 2 · TUE
Day 2 — Chain Rule & Logarithmic Functions
Sections 3–4 — Chain Rule & Logarithmic Functions
Chain Rule[f(g(x))]' = f'(g(x))·g'(x) — peel outer, multiply by inner
Log basicd/dx(log x) = 1/x
Log composited/dx[log f(x)] = f'(x)/f(x)
TipSimplify using log laws BEFORE differentiating
✎Concept Board — Day 2Click to open · Draw diagrams & explain concepts▶
Q7
\((3x^2+5)^5\) — Chain Rule (2M)
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Q8
\(\sqrt{2x+3}\) — Chain Rule (2M)
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Q9
\(\sin(ax+b)\) — Chain Rule (2M)
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Q10
\(\log(x^2+3x)\) — Log of a function (2M)
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Q11
\(\log\sqrt{\sin x}\) — Log + Simplify (2M)
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Q12
\(\log\!\left(\dfrac{x-1}{x+1}\right)\) — Log subtraction (3M)
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Q13
\(\log\!\left(\dfrac{x^3}{x^2-1}\right)\) — Log simplification (3M)
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📎 Assignments — Day 2 (Optional · Viewable by students)
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Day 2 — Video Resources
Exit Ticket
1State the Chain Rule.
2Find d/dx[log(sin x)].
Key Formulas — Day 2
•Chain: [f(g(x))]' = f'(g(x))·g'(x)
•Log: [log f(x)]' = f'(x)/f(x)
•Simplify logs first!
DAY 3 · WED
Day 3 — Exponential & Inverse Trig (ITF Basics)
Sections 5–6 — Exponential & ITF Basics
e^f(x)d/dx(e^f(x)) = e^f(x)·f'(x)
aˣd/dx(aˣ) = aˣ·log a
sin⁻¹d/dx(sin⁻¹x) = 1/√(1−x²)
cos⁻¹d/dx(cos⁻¹x) = −1/√(1−x²)
tan⁻¹d/dx(tan⁻¹x) = 1/(1+x²)
✎Concept Board — Day 3Click to open · Draw diagrams & explain concepts▶
Q14
\(e^{3x^2-4x+5}\) — Exponential Chain Rule (2M)
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Q15
\(5^x\) — Exponential base \(a\) (1M)
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Q16
\(e^{\sin 2x}\) — Exponential + trig chain (2M)
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Q17
\(e^{3\log x}\) — Simplify then differentiate (2M)
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Q18
\(\sin^{-1}(2x)\) — ITF Basic (2M)
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Q19
\(\tan^{-1}(x^2)\) — ITF Basic + Chain (2M)
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📎 Assignments — Day 3 (Optional · Viewable by students)
A1
A2
A3
Day 3 — Video Resources
Exit Ticket
1State d/dx(e^sinx).
2State d/dx(sin⁻¹x).
Key Formulas — Day 3
•e^f(x): mult by f'(x)
•(sin⁻¹x)' = 1/√(1−x²)
•(tan⁻¹x)' = 1/(1+x²)
DAY 4 · THU
Day 4 — ITF by Trigonometric Substitution
Section 7 — ITF by Trigonometric Substitution
sin⁻¹(2x√(1−x²))→ x=sinθ → sin2θ → 2sin⁻¹x
tan⁻¹(2x/(1−x²))→ x=tanθ → tan2θ → 2tan⁻¹x
cos⁻¹((1−x²)/(1+x²))→ x=tanθ → cos2θ → 2tan⁻¹x
KeyCheck domain before simplifying sin⁻¹(sinθ)=θ
✎Concept Board — Day 4Click to open · Draw diagrams & explain concepts▶
Q20
\(\sin^{-1}(2x\sqrt{1-x^2})\) — Substitute \(x=\sin\theta\) (3M)
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Q21
\(\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)\) — Substitute \(x=\tan\theta\) (3M)
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Q22
\(\tan^{-1}\!\sqrt{\dfrac{1-x}{1+x}}\) — Substitute \(x=\cos 2\theta\) (3M)
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Q23
\(\sin^{-1}\!\left(x\sqrt{1-x^2}+\sqrt{1-x^2}\cdot x\right)\) — Simplify (3M)
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Q24
\(\cos^{-1}(2x^2-1)\) — Substitute \(x=\cos\theta\) (3M)
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Q25
\(\cos^{-1}\!\left(\dfrac{1-x^2}{1+x^2}\right)\) — Substitute \(x=\tan\theta\) (3M)
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Q26
\(\tan^{-1}\!\left(\dfrac{3x-x^3}{1-3x^2}\right)\) — Substitute \(x=\tan\theta\) (3M)
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Q27
\(\sin^{-1}(3x-4x^3)\) — Substitute \(x=\sin\theta\) (3M)
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Q28
\(\cos^{-1}(4x^3-3x)\) — Substitute \(x=\cos\theta\) (3M)
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Q29
\(\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)\) — Substitute \(x=\tan\theta\) (3M)
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📎 Assignments — Day 4 (Optional · Viewable by students)
A1
A2
A3
Day 4 — Video Resources
Exit Ticket
1For sin⁻¹(2x√(1−x²)), which substitution?
2After x=tanθ in tan⁻¹(2x/(1−x²)), result?
Substitution Guide
•sin⁻¹ types → x=sinθ
•tan⁻¹ types → x=tanθ
•cos⁻¹ types → x=cosθ or tanθ
•Triple-angle → same substitution
DAY 5 · FRI
Day 5 — Implicit Differentiation & ∞-Series
Section 8 — Implicit Differentiation & ∞-Series
Key RuleDifferentiate both sides w.r.t. x — every y-term gets dy/dx factor
CollectGroup all dy/dx terms, factorise, isolate
∞-Seriesy = √(f(x)+y) → square → y²=f(x)+y → differentiate
✎Concept Board — Day 5Click to open · Draw diagrams & explain concepts▶
Q30
\(x^2+y^2=a^2\), find \(dy/dx\) (2M)
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Q31
\(x^3+y^3=3axy\), find \(dy/dx\) (3M)
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Q32
\(\sin(x+y)=k\), show \(dy/dx=-1\) (3M)
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Q33
\(\tan x+\tan y=c\), find \(dy/dx\) (2M)
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Q34
\(y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\cdots\infty}}}\), find \(dy/dx\) (3M)
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Q35
\(y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\cdots\infty}}}\), find \(dy/dx\) (3M)
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📎 Assignments — Day 5 (Optional · Viewable by students)
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Day 5 — Video Resources
Exit Ticket
1Differentiate x²+y²=r² implicitly.
2If y=√(sinx+y), squaring gives?
Key Steps
•Diff both sides w.r.t. x
•Every y-term → dy/dx factor
•∞-series: y²=f(x)+y then diff
•Factorise dy/dx
DAY 6 · SAT
Day 6 — Parametric & Logarithmic Differentiation
Sections 9–10 — Parametric & Log Differentiation
Parametricdy/dx = (dy/dt) ÷ (dx/dt)
Log DiffTake log y = ..., diff: (1/y)·dy/dx = ..., multiply by y
Complex fnUse log diff when base AND exponent both have x
✎Concept Board — Day 6Click to open · Draw diagrams & explain concepts▶
Q36
\(x=a\cos t,\; y=b\sin t\), find \(dy/dx\) (2M)
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Q37
\(x=a(t-\sin t),\; y=a(1-\cos t)\), find \(dy/dx\) (3M)
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Q38
\(x=a\cos t,\; y=a\sin t\), find \(dy/dx\) (2M)
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Q39
\(y=x^{\sin x}\), find \(dy/dx\) using log differentiation (3M)
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Q40
\(y=x^{\log x}\), find \(dy/dx\) (3M)
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Q41
\(y=\dfrac{(x-1)^3(x-2)^4}{(x-3)^5(x-4)^6}\), find \(dy/dx\) using log differentiation (4M)
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📎 Assignments — Day 6 (Optional · Viewable by students)
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Day 6 — Video Resources
Exit Ticket
1State parametric diff formula.
2For y=x^sinx, what is log y?
Key Formulas
•dy/dx = (dy/dt)÷(dx/dt)
•Log diff: log y → (1/y)·dy/dx
•Multiply by y at end
DAY 7 · SUN
Day 7 — Second Order Derivatives & ISC Proofs
Section 11 — Second Order Derivatives & ISC Proofs
Definitiony₂ = d/dx(y₁) where y₁ = dy/dx
Notationd²y/dx² = y₂ = D²y = f''(x)
ISC PatternFind y₁, y₂ → substitute in LHS → show = 0
✎Concept Board — Day 7Click to open · Draw diagrams & explain concepts▶
Q42
If \(y=e^{ax}\sin bx\), prove \(y_2-2ay_1+(a^2+b^2)y=0\) (4M)
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Q43
If \(y=e^{ax}\cos bx\), prove \(y_2-2ay_1+(a^2+b^2)y=0\) (4M)
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Q44
If \(y=Ae^{mx}+Be^{nx}\), prove \(y_2-(m+n)y_1+mny=0\) (4M)
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Q45
If \(y=(\sin^{-1}x)^2\), prove \((1-x^2)y_2-xy_1-2=0\) (4M)
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Q46
If \(y=\sin(m\sin^{-1}x)\), prove \((1-x^2)y_2-xy_1+m^2y=0\) [CBSE 2024] (4M)
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Q47
If \(y=\log(x+\sqrt{x^2+1})\), prove \((1+x^2)y_2+xy_1=0\) (4M)
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Q48
\(x=a(\theta-\sin\theta),\; y=a(1-\cos\theta)\), find \(d^2y/dx^2\) (4M)
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Q49
If \(y=e^{ax}\cos bx\), show \(y_2-2ay_1+(a^2+b^2)y=0\) [CBSE 2019C] (4M)
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📎 Assignments — Day 7 (Optional · Viewable by students)
A1
A2
A3
Day 7 — Video Resources
Exit Ticket
1State definition of d²y/dx².
2If y=e^ax, find d²y/dx².
3Pattern of ISC proof questions?
ISC Proof Strategy
•Find y₁=dy/dx
•Find y₂=d(y₁)/dx
•Substitute in LHS
•Show LHS=0
🎬 Differentiation — Interactive Animation