CBSE · Class 12 · Mathematics · Chapter 6

Application of
Derivatives

Complete chapter resources for CBSE Class 12 Maths — rate of change, increasing and decreasing functions, tangents and normals, maxima and minima, with sample questions and previous year board papers.

5Topics
10–12Board marks
8Sample questions
3PYQ included
Sign Up Free & Generate → See sample questions ↓

Free for independent teachers · No credit card required

Key Formulas — Chapter 6
  • Rate of change: dy/dx = rate of change of y w.r.t. x
  • Increasing: f'(x) > 0 on (a, b) → f increasing
  • Slope of tangent: m = dy/dx|(x₁,y₁)
  • Normal slope: m_n = −1 / (dy/dx)
  • 2nd derivative test: f''(c) < 0 → local max; f''(c) > 0 → local min
  • Absolute extrema: evaluate f at critical points + endpoints
Chapter Overview

What this chapter covers

Chapter 6 of CBSE Class 12 Mathematics builds on differentiation (Chapter 5) and explores the diverse ways derivatives describe real-world behaviour. The derivative dy/dx measures the instantaneous rate of change of y with respect to x — whether the variable represents position, volume, temperature, or profit. This chapter shows students how to read that information geometrically and algebraically.

A central theme is using the sign of f'(x) to determine where a function is increasing or decreasing, and using f'(x) = 0 (critical points) together with the first or second derivative test to classify local maxima and minima. The chapter also covers equations of tangents and normals to curves — a topic that bridges algebra and geometry and appears in almost every CBSE board paper.

The chapter closes with maxima and minima in word problems — optimising area, volume, cost, or profit given a constraint. These problems require students to set up the objective function, differentiate, find critical points, verify using derivative tests, and check endpoints on closed intervals. They typically carry 5 marks and are the highest-scoring opportunity in this chapter.

Topics

What's inside Chapter 6

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Rate of Change of Quantities
Using dy/dx to find how one quantity changes with respect to another — area expanding, volume filling, distance with speed. Includes related-rates chain-rule problems.
Topic 2
Increasing and Decreasing Functions
A function f is increasing on (a, b) if f'(x) > 0 and decreasing if f'(x) < 0. Finding intervals of increase/decrease by solving f'(x) = 0 and testing signs in each interval.
Topic 3
Tangents and Normals
Slope of tangent = dy/dx at the point. Tangent equation: y − y₁ = m(x − x₁). Normal is perpendicular: slope = −1/m. Special cases: horizontal tangent (m = 0), vertical tangent (m undefined).
Topic 4
Maxima and Minima — Local
First derivative test: sign change of f'(x) around a critical point. Second derivative test: f''(c) < 0 is local max, f''(c) > 0 is local min, f''(c) = 0 is inconclusive.
Topic 5
Absolute Maxima and Minima
On a closed interval [a, b]: evaluate f at all critical points and both endpoints; the largest value is the absolute maximum, smallest is the absolute minimum. Applies to real-world optimisation problems.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 5 · Continuity & Differentiability
Derivatives of standard functions, chain rule, implicit differentiation
Ch 13 (Class 11) · Limits
First-principle definition of the derivative, limit-based slope of a curve
Chapter 6 Application of
Derivatives
Leads to
Ch 7 · Integrals
Integration as the reverse of differentiation; area under curves
Ch 8 · Application of Integrals
Area between curves — directly uses increasing/decreasing analysis from Ch 6
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Assertion–Reason 1 1–2 Nature of critical point, sign of derivative, or slope of tangent at a given point
Very Short Answer 2 1 Equation of tangent/normal, rate of change, or intervals of increase/decrease
Short Answer 3 1 Find local maxima/minima using first or second derivative test
Long Answer / Optimisation 5 1 Real-world optimisation — box, cylinder, window, fencing, profit/cost
Total (approximate) 10–12 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 6. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The slope of the tangent to the curve y = x³ − 3x at x = 2 is: (a) 9 (b) 3 (c) 6 (d) 12
Q2 Easy 2 marks Short Answer
The radius of a circle is increasing at the rate of 0.5 cm/s. Find the rate of increase of its area when the radius is 8 cm.
Q3 Medium 2 marks Short Answer
Find the equation of the tangent and the normal to the curve y = x² − 4x + 5 at the point (2, 1).
Q4 Medium 3 marks Short Answer
Find the intervals on which f(x) = 2x³ − 9x² + 12x − 5 is (i) strictly increasing and (ii) strictly decreasing.
Q5 Medium 3 marks Short Answer
Find all local maxima and local minima of f(x) = x³ − 6x² + 9x + 15 using the second derivative test.
Q6 Hard 4 marks Word Problem
A ladder 5 m long is leaning against a wall. The bottom of the ladder is being pulled away from the wall at 1 m/s. Find the rate at which the top of the ladder is sliding down the wall when the bottom is 3 m from the wall.
Q7 Hard 5 marks Word Problem
A rectangular sheet of tin 45 cm × 24 cm is to be made into a box without a top by cutting equal squares from each corner and folding up the flaps. Find the side of the square to be cut so that the volume of the box is maximum. Also find the maximum volume.
Q8 Hard 5 marks Word Problem
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Generate a full paper with answer key →

MarksZen AI creates a complete question paper with answer key in under 2 minutes.

Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Application of Derivatives chapter.

Board 20222 marks
Find the intervals in which the function f(x) = 2x³ − 3x is strictly increasing and strictly decreasing. (CBSE All India 2022)
Board 20233 marks
Find the local maxima and local minima, if any, of the function f(x) = sin x + cos x, 0 < x < π/2. Also find the local maximum and local minimum values. (CBSE Delhi 2023)
Board 20205 marks
An open box with a square base is to be made from a given quantity of cardboard of area c² square units. Show that the maximum volume of the box is c³ / (6√3) cubic units. (CBSE 2020)
For Teachers

Create a board-aligned
question paper in 2 minutes.

Pick chapter, set the question-type mix and total marks — MarksZen AI generates the full paper with answer key. CBSE, ICSE, and all State Boards supported.

  • All 5 topics of this chapter
  • MCQ + short answer + optimisation problems
  • Answer key included
  • PDF export ready
Sign Up Free & Generate →
FAQ

Questions teachers ask

How many marks does Application of Derivatives carry in the CBSE Class 12 board exam? +
Application of Derivatives typically carries 10–12 marks in the CBSE Class 12 Mathematics board exam. Questions span all types — 1-mark MCQs on nature of critical points, 2-mark questions on tangents/normals, and 5-mark long-answer optimisation word problems. This is one of the highest-weightage chapters in the Calculus unit.
What is the first derivative test and when should students use it in board exams? +
The first derivative test determines whether a critical point (where f'(x) = 0 or f'(x) is undefined) is a local maximum, local minimum, or neither. If f'(x) changes from positive to negative at x = c, then f has a local maximum there; if it changes from negative to positive, a local minimum. Use it when the second derivative is zero or difficult to compute — the examiner awards full marks for either test, provided working is shown.
How do you find the equation of the tangent and normal to a curve at a given point? +
Find the slope of the tangent by evaluating dy/dx (the derivative of the curve's equation) at the given point (x₁, y₁). The tangent equation is: y − y₁ = m(x − x₁), where m = dy/dx at (x₁, y₁). The normal is perpendicular to the tangent, so its slope is −1/m, giving: y − y₁ = (−1/m)(x − x₁). These are 2–3 mark questions in board exams and almost always appear.
What is the difference between absolute maximum/minimum and local maximum/minimum in CBSE Class 12? +
A local (relative) maximum is a point where the function value is greater than all nearby values; an absolute maximum is the greatest value over the entire domain (or closed interval). For closed-interval optimisation problems in board exams, students must evaluate f at all critical points AND both endpoints, then compare — the largest is the absolute maximum, the smallest is the absolute minimum. Forgetting to check endpoints is a common source of lost marks.
How do I generate a custom question paper for Application of Derivatives using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 6 (Application of Derivatives), set your preferred question-type mix (MCQ, short answer, word problem) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.