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Mathematics — Class 12 — L26: Differentiation

NIOS Code 311 · Module 8 · Calculus

Notes extracted from NIOS Mathematics Course (311), Lesson 26 — Differentiation (ilovepdf_merged (6).pdf). Content covers sections 26.1–26.x.

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Overview — Differentiation (L26)

The derivative measures instantaneous rate of change and the slope of the tangent to y=f(x).

f′(x) = lim_(h→0) [f(x+h)−f(x)] / h

h≠0 · provided the limit exists

Derivative as slope of the tangent

26.1 Standard derivatives

d/dx (xⁿ) = n xⁿ⁻¹ · d/dx (c) = 0 · d/dx (eˣ)=eˣ · d/dx (ln x)=1/x

Power, constant, exponential, log

Sum/difference: (u±v)′ = u′±v′.
Product: (uv)′ = u′v + uv′.
Quotient: (u/v)′ = (u′v − uv′)/v².
Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).

26.2 Geometry

Equation of tangent at x=a: y − f(a) = f′(a)(x − a). Normal has slope −1/f′(a) when f′(a)≠0.

Differentiability: Differentiable ⇒ continuous; converse false (e.g. |x| at 0).

MCQ Quiz — L26 Differentiation

0 / 10 correct

Flashcards — L26

1 / 16

Golden Rules — L26 Differentiation

Most exam-important points from this chapter:

Know the definitions of L26

Start every answer with the key definition or standard form from Differentiation.

Write the formula first

Copy the formula, then substitute values; never jump to the number alone.

Watch conditions

Check domain, quadrant, non-zero denominators, and applicability of the theorem.

Master result 1

Memorise and apply: necessarily zero. The limit,

Master result 2

Memorise and apply: is equivalently represented by dy .

f'(x)=lim h→0 [f(x+h)−f(x)]/h

Power rule

Product rule

Quotient rule

Chain rule

(c)'=0

1. Formulas & Definitions

Full Ch 26 — Differentiation study guide: definitions, derivations, why each formula works, history, deep understanding, pencil diagrams, step-by-step use, and 4-level solved examples.

d/dx (xⁿ) = n x^{n−1}

Definition: Power rule for real n (with domain care).

Derivation

From first principles for positive integers; extended by algebra/log.

Variables

n constant

Why it works

Local linear rate of change of power functions.

Historical context

Newton–Leibniz calculus core rule.

Deep understanding

Works for n negative/fractional on suitable domains.

2. Diagrams & Visuals

Pencil sketch · labelled · step-by-step breakdown below

Identify power n
Bring n down
Reduce exponent by 1
Simplify

3. Solved Examples

Basic

Q: d/dx x⁵

Solution: 5x⁴

Answer: 5x⁴

Intermediate

Q: d/dx √x

Solution: (1/2)x^{−1/2}

Answer: 1/(2√x)

Advanced

Q: d/dx (1/x³)

Solution: −3/x⁴

Answer: −3 x^{−4}

Exam

Q: d/dx (5x³−2x+1)

Solution: 15x²−2

Answer: 15x² − 2

(uv)' = u'v + uv'

Definition: Product rule.

Derivation

Expand [u(x+h)v(x+h)−u v]/h and take limit.

Variables

u,v differentiable

Why it works

Both factors can change; total rate is sum of each changing alone.

Historical context

Leibniz product rule.

Deep understanding

Extends to three factors: u'vw+uv'w+uvw'.

2. Diagrams & Visuals

Pencil sketch · labelled · step-by-step breakdown below

Differentiate first keep second
Plus keep first differentiate second
Simplify
Factor if useful

3. Solved Examples

Basic

Q: (x·x)'

Solution: 1·x+x·1=2x

Answer: 2x

Intermediate

Q: (x² sin x)'

Solution: 2x sin x + x² cos x

Answer: 2x sin x + x² cos x

Advanced

Q: (eˣ x)'

Solution: eˣ x + eˣ

Answer: eˣ(x+1)

Exam

Q: State product rule.

Solution: (uv)'=u'v+uv'

Answer: u'v + uv'

d/dx f(g(x)) = f'(g(x)) g'(x)

Definition: Chain rule for composite functions.

Derivation

Derivative of outer at inner times derivative of inner.

Variables

f outer · g inner

Why it works

Rates multiply along a chain of dependence.

Historical context

Key theorem of differential calculus.

Deep understanding

Leibniz notation: dy/dx = (dy/du)(du/dx).

2. Diagrams & Visuals

Pencil sketch · labelled · step-by-step breakdown below

Identify outer and inner
Diff outer w.r.t inner
Multiply by inner derivative
Simplify

3. Solved Examples

Basic

Q: d/dx (x²+1)³

Solution: 3(x²+1)²·2x

Answer: 6x(x²+1)²

Intermediate

Q: d/dx sin(5x)

Solution: 5 cos 5x

Answer: 5 cos 5x

Advanced

Q: d/dx e^{x²}

Solution: e^{x²}·2x

Answer: 2x e^{x²}

Exam

Q: Chain rule statement.

Solution: dy/dx=(dy/du)(du/dx)

Answer: Outer' × inner'

5. Special Features & Extras

Complete study guide — exam tips, common mistakes, memory aids, and quick reference.

Exam Tips & Tricks

Write chain rule with substitution u=g(x) if stuck.
Product vs chain: product is two factors multiplied, chain is composition.
Simplify before differentiating if easy.

Common Student Mistakes

Forgetting chain factor
Using product rule on composition
Power rule on (sin x)ⁿ without chain

Memory Aids & Mnemonics

Product: each factor takes a turn to differentiate.

Chain: outer' × inner'.

Which Formula When?

Power of x → power rule
Product → product rule
f(g(x)) → chain

Quick reference box

• nx^{n−1} · u'v+uv' · f'(g)g'

PYQ — Previous Year Questions

English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.

L26 — Differentiation

5 question(s) · Sources: Board-style chain, Board-style product, Board-style tangent, Sample QP 2024

PYQ1. If y = xⁿ, then dy/dx equals:

(A) n xⁿ⁺¹
(B) n xⁿ⁻¹
(C) xⁿ / n
(D) n xⁿ

1 mark(s) · MCQ · Sample QP 2024 · Q13(i)

Model Answer

1) n xⁿ⁻¹

Explanation

1) Power rule: d/dx (xⁿ) = n xⁿ⁻¹ (n constant).

PYQ2. If y = x² sin x, find dy/dx.

2 mark(s) · SA · Board-style product

Model Answer

1) 2x sin x + x² cos x

Explanation

1) Product rule: (x²)′sin x + x²(cos x) = 2x sin x + x² cos x.

PYQ3. Find d/dx (2x+3)⁴.

1 mark(s) · SA · Board-style chain

Model Answer

1) 8(2x+3)³

Explanation

1) Chain rule: 4(2x+3)³ · 2 = 8(2x+3)³.

PYQ4. Find the equation of the tangent to y=x² at x=2.

2 mark(s) · SA · Board-style tangent

Model Answer

1) y = 4x − 4

Explanation

1) Point (2,4); m=2x=

4. y−4=4(x−2) ⇒ y=4x−4.

PYQ5. Write ∫ sec²(mx) dx (m≠0).

1 mark(s) · SA · Sample QP 2024 · Q13(ii) OR concept

Model Answer

1) (1/m) tan(mx) + C

Explanation

1) d/dx tan(mx)=m sec²(mx) ⇒ ∫sec²(mx)dx = tan(mx)/m + C.

Problem Solving — L26 Differentiation

10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.

Question 1 of 10Power

Differentiate y = x⁵ − 3x² + 7.

(xⁿ)′=n xⁿ⁻¹

1) dy/dx = 5x⁴ − 6x + 0 = 5x⁴ − 6x.

Answer: 5x⁴ − 6x

Formula used

(xⁿ)′=n xⁿ⁻¹

Textbook formal language

Termwise power rule; constant differentiates to 0.

Easy language (same calculation)

Bring power down, reduce power by 1.

Why this formula

Linearity of differentiation.

Exam tip

Constant 7 vanishes.

Common mistakes

Leaving +7
5x⁵

Question 2 of 10Product

If y = x² sin x, find dy/dx.

(uv)′=u′v+uv′

1) u=x², v=sin x ⇒ y′=2x sin x + x² cos x.

Answer: 2x sin x + x² cos x

Formula used

(uv)′=u′v+uv′

Textbook formal language

Product rule applied to x² and sin x.

Easy language (same calculation)

Differentiate first times second plus first times derivative of second.

Why this formula

Can factor x: x(2 sin x + x cos x).

Exam tip

Do not differentiate only one factor.

Common mistakes

2x cos x only
x² cos x only

Question 3 of 10Quotient

Differentiate y = (x+1)/(x−1), x≠1.

(u/v)′=(u′v−uv′)/v²

1) u=x+1,v=x−1 ⇒ y′=[1·(x−1)−(x+1)·1]/(x−1)² = (x−1−x−1)/(x−1)² = −2/(x−1)².

Answer: −2/(x−1)²

Formula used

(u/v)′=(u′v−uv′)/v²

Textbook formal language

Quotient rule simplifies to −2/(x−1)².

Easy language (same calculation)

Bottom times 1 minus top times 1, over bottom squared.

Why this formula

Defined for x≠1.

Exam tip

Numerator becomes −2, not 0.

Common mistakes

2/(x−1)²
Forgetting square on denominator

Question 4 of 10Chain

Find d/dx (2x+3)⁴.

d/dx f(g)=f′(g)g′

1) Let u=2x+3; d/dx u⁴ = 4u³ · 2 = 8(2x+3)³.

Answer: 8(2x+3)³

Formula used

d/dx f(g)=f′(g)g′

Textbook formal language

Chain rule: outer power times inner derivative 2.

Easy language (same calculation)

4(something)³ times 2.

Why this formula

General (ax+b)ⁿ → n(ax+b)ⁿ⁻¹ · a.

Exam tip

Do not omit the factor 2.

Common mistakes

4(2x+3)³
8(2x+3)⁴

Question 5 of 10exp log

Differentiate y = eˣ + ln x (x>0).

(eˣ)′=eˣ, (ln x)′=1/x

1) y′ = eˣ + 1/x.

Answer: eˣ + 1/x

Formula used

(eˣ)′=eˣ, (ln x)′=1/x

Textbook formal language

Sum of standard derivatives.

Easy language (same calculation)

eˣ stays eˣ; ln becomes 1/x.

Why this formula

Domain x>0 for real ln.

Exam tip

Do not write 1/eˣ.

Common mistakes

eˣ only
x + 1/x

Question 6 of 10Tangent

Find the equation of the tangent to y=x² at x=2.

y−y₀=m(x−x₀), m=f′(x₀)

1) y(2)=4; m=f′(x)=2x ⇒ m=4.

2) Tangent: y−4=4(x−2) ⇒ y=4x−4.

Answer: y = 4x − 4

Formula used

y−y₀=m(x−x₀), m=f′(x₀)

Textbook formal language

Point-slope form with derivative as slope.

Easy language (same calculation)

At x=2 point is (2,4) and slope 4.

Why this formula

Normal would have slope −1/4.

Exam tip

Use the point on the curve, not (2,0).

Common mistakes

y=2x
y=4x

Question 7 of 10Power

Differentiate y = x⁷ − 4x³ + 2.

(xⁿ)′ = n x^{n−1}

1) y′ = 7x⁶ − 12x².

Answer: 7x⁶ − 12x²

Formula used

(xⁿ)′ = n x^{n−1}

Textbook formal language

Term-by-term power rule; constant derivative zero.

Easy language (same calculation)

Bring powers down.

Why this formula

2 vanishes.

Exam tip

Check each algebraic step carefully.

Common mistakes

7x⁷
Keeping +2

Question 8 of 10Product

Differentiate y = x² cos x.

(uv)′=u′v+uv′

1) y′ = 2x cos x + x² (−sin x) = 2x cos x − x² sin x.

Answer: 2x cos x − x² sin x

Formula used

(uv)′=u′v+uv′

Textbook formal language

Product rule with u=x², v=cos x.

Easy language (same calculation)

Each factor takes a turn to differentiate.

Why this formula

Derivative of cos is −sin.

Exam tip

Check each algebraic step carefully.

Common mistakes

2x cos x only
+ x² sin x

Question 9 of 10Chain

Differentiate y = (3x + 1)⁵.

f(g(x))′ = f′(g) g′

1) y′ = 5(3x+1)⁴ · 3 = 15(3x+1)⁴.

Answer: 15(3x + 1)⁴

Formula used

f(g(x))′ = f′(g) g′

Textbook formal language

Chain rule: outer power, inner linear.

Easy language (same calculation)

Power 5 comes down; multiply by 3.

Why this formula

Do not forget the inner derivative.

Exam tip

Check each algebraic step carefully.

Common mistakes

5(3x+1)⁴
(3x+1)⁵

Question 10 of 10Quotient

Differentiate y = (x)/(x+1).

(u/v)′=(u′v−uv′)/v²

1) u=x, v=x+1; u′=1, v′=1.

2) y′ = [(1)(x+1) − x(1)]/(x+1)² = 1/(x+1)².

Answer: 1/(x+1)²

Formula used

(u/v)′=(u′v−uv′)/v²

Textbook formal language

Quotient rule simplifies neatly.

Easy language (same calculation)

Numerator becomes 1.

Why this formula

Domain x≠−1.

Exam tip

Check each algebraic step carefully.

Common mistakes

1/(x+1)
(x+1−x)/(x+1)