Differentiation Notes & Questions | Class 12 Mathematics | Drishti OS

Differentiation

What you'll learn

The derivative as rate of change and slope of tangent — definition f′(x) = lim(h→0) [f(x+h)−f(x)]/h.
Standard derivatives: power, trigonometric, exponential, logarithmic, and inverse trig.
Rules: product, quotient, chain (composition), and implicit differentiation.
To find tangents, normals, and interpret increasing/decreasing intervals via f′ sign.
Second derivative for concavity and points of inflection — JEE application problems.

Key concepts

Level 1 — Foundations

Verbal: Differentiation finds how fast a quantity changes. Geometrically, f′(a) is the slope of the tangent to y = f(x) at x = a.

First principle: f′(x) = lim(h→0) [f(x+h) − f(x)]/h.

Basic derivatives (NCERT table):

f(x)	f′(x)
xⁿ	nxⁿ⁻¹
sin x	cos x
cos x	−sin x
eˣ	eˣ
ln x	1/x
aˣ	aˣ ln a

Differentiability ⟹ continuity (converse false: |x| at 0 is continuous but not differentiable).

Level 2 — JEE / NEET depth

Rules:

Sum: (u ± v)′ = u′ ± v′
Product: (uv)′ = u′v + uv′
Quotient: (u/v)′ = (u′v − uv′)/v², v ≠ 0
Chain: If y = f(g(x)), then dy/dx = f′(g(x))·g′(x)

Parametric: If x = φ(t), y = ψ(t), then dy/dx = (dy/dt)/(dx/dt).

Logarithmic differentiation: For y = f(x)^g(x), take ln both sides then differentiate — essential for variable exponents.

Applications:

Problem	Method
Tangent at (a, f(a))	y − f(a) = f′(a)(x − a)
Normal slope	−1/f′(a) if f′(a) ≠ 0
Max/min	f′ = 0, sign chart or second derivative test
Related rates	Differentiate equation w.r.t. time t

Mean Value Theorem (Rolle/Lagrange): If f is continuous on [a,b], differentiable on (a,b), then ∃ c with f′(c) = [f(b)−f(a)]/(b−a).

Worked example

Differentiate using chain rule

y = sin(3x² + 1).

Step 1 — Outer function sin u, inner u = 3x² + 1.
Step 2 — dy/dx = cos u · d(3x²+1)/dx = cos(3x²+1) · 6x.
Step 3 — Result: dy/dx = 6x cos(3x² + 1).

Find equation of tangent to y = x³ − 3x at x = 2

Step 1 — Point: f(2) = 8 − 6 = 2 → (2, 2).
Step 2 — f′(x) = 3x² − 3 → f′(2) = 12 − 3 = 9.
Step 3 — Tangent: y − 2 = 9(x − 2).
Step 4 — Simplify: y = 9x − 16.

Common mistakes

Mistake	Why it happens	Fix
Forgetting chain rule inner derivative	Differentiating outer only	Multiply by derivative of inside
Quotient rule numerator order	Swapping u′v and uv′	Remember: (u′v − uv′)/v²
Differentiating sin⁻¹ without domain	Missing √(1−x²) denominator	d/dx sin⁻¹x = 1/√(1−x²),
Assuming differentiable ⟹ continuous fails conversely	One-way implication reversed

Quick check

Differentiate y = x² e^(3x).
Find f′(π/4) if f(x) = tan x.
Where is f(x) = |x² − 4| non-differentiable?
If s = t³ − 6t, find velocity at t = 2.
Stretch: Use logarithmic differentiation on y = x^x for x > 0.

NCERT Chapter 5 link: Differentiation rules accumulate — maintain a single revision sheet combining chain, product, quotient, and standard formulas. NCERT Exercise 5.5 on logarithmic differentiation appears regularly in boards.

Exam connections: JEE combines related rates (ladder problems, balloon expansion) with implicit differentiation (circle x²+y² = r²). Normal and tangent line questions require both point and slope — find dy/dx first, then apply line formula. Increasing/decreasing intervals use open intervals where f′ > 0 or f′ < 0.

Study strategy: When differentiating nested functions, count layers before starting — write u, v substitutions in margin. For max/min word problems, define variable, form constraint equation, differentiate once, test critical points with second derivative or sign chart.

Study workflow and exam preparation

When studying Differentiation within Calculus, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Coordinate geometry and vectors often combine with matrices; calculus links to physics kinematics problems.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Mathematics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Differentiation.

Differentiation