This page hosts the Application of Derivatives Class 12 NCERT Solutions, presenting it for Exercise 6.2 of Class 12 Mathematics Chapter 6 Application of Derivatives. Each solution in these notes explicitly names the theorem or formula applied, then proceeds line-by-line to the final answer. Aligned to the 2026-27 NCERT syllabus.
- CBSE Weightage: 5-7 marks from Application of Derivatives
- JEE Main Coverage: 3-5% of the calculus segment
- Exercise 6.2 Problems: 19 questions (mix of long, short and MCQ)
Student Pulse - Application of Derivatives Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
The solutions below follow the 2026-27 NCERT syllabus and demonstrate every concept step by step. Students preparing for board exams as well as competitive tests will find each problem solved in the simplest possible form, with the increasing-decreasing test, the first derivative sign analysis, and interval-based reasoning made explicit. Collegedunia's solutions emphasise method clarity over shortcuts.
NCERT Solutions for Class 12 Maths Chapter 6 Exercise 6.2 - Topics Covered
The this Class 12 page address this in the same order as the NCERT textbook.
Exercise 6.2 focuses on a single high-yield idea: classifying a function as increasing, decreasing, strictly increasing, or strictly decreasing using the sign of its first derivative. The table below maps the problem types in the exercise to the underlying concept tested.
| Problem Type | Concept Tested | Question Numbers |
|---|---|---|
| Show f(x) is strictly increasing/decreasing on R | Sign of f'(x) on the entire real line | Q1, Q2, Q3 |
| Find intervals of monotonicity | Critical points + sign chart of f'(x) | Q5, Q6, Q7, Q8 |
| Trigonometric monotonicity | Periodic behaviour of sin x, cos x derivatives | Q9, Q10, Q11 |
| Logarithmic and polynomial | Derivatives of log, exponential and powers | Q12, Q13, Q14 |
| MCQ on monotonic intervals | Quick critical-point identification | Q18, Q19 |
Application of Derivatives Ex 6 2 Video Walkthrough
Source: Magnet Brains on YouTube
How the Application of Derivatives Class 12 NCERT Solutions on the Application of Derivatives Class 12 NCERT Solutions Help You
The the resource address this in the same order as the NCERT textbook.
The increasing-decreasing function test trips up students who mix up f'(x) > 0 implies strictly increasing with the weaker monotonic version. Our solutions distinguish these clearly. Each problem opens with the derivative computation, moves through the sign analysis of f'(x), and closes with the explicit interval declaration. You also get:
- Sign charts drawn for every critical-point split
- Domain checks before applying the derivative test
- Common pitfalls flagged where boundary points need special handling
- Quick verification using sample values inside each candidate interval
Key Formulae Used in Exercise 6.2
The chapter notes address this in the same order as the NCERT textbook.
Exercise 6.2 leans on a small but powerful set of derivative rules. Memorise the table below; every problem in the exercise reduces to applying one or two of these.
| Function | Derivative | Sign Behaviour |
|---|---|---|
| f(x) = xn | nxn-1 | Depends on n and sign of x |
| f(x) = sin x | cos x | Positive on (-π/2, π/2) |
| f(x) = cos x | -sin x | Negative on (0, π) |
| f(x) = ex | ex | Always positive |
| f(x) = log x | 1/x | Positive on (0, ∞) |
Full formula sheet: Class 12 Maths Chapter 6 Formula Sheet
Solved Example from Ex 6.2 - Strictly Increasing Function
Show that the function f(x) = 3x + 17 is strictly increasing on R.
Step 1. Compute the derivative: f'(x) = 3 .
Step 2. Since f'(x) = 3 > 0 for every real number x, the function is strictly increasing on the entire real line R. Strictly positive derivative on an interval implies strictly increasing on that interval.
Common Mistakes in Class 12 Maths Exercise 6.2
The the PDF are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
Many students lose marks on monotonicity questions because of avoidable algebraic slips, not because they misunderstand the concept. Here are the most frequent errors observed in CBSE Board answer scripts.
- Forgetting to check the domain of the function before computing f'(x)
- Confusing "non-decreasing" with "strictly increasing"
- Missing the boundary points when an interval is closed at one end
- Not testing a sample point inside each candidate interval to confirm the sign
Related Resources
- NCERT Solutions Class 12 Maths Chapter 6 Exercise 6.1
- NCERT Solutions Class 12 Maths Chapter 6 Exercise 6.3
- NCERT Solutions Class 12 Maths Chapter 6 Miscellaneous Exercise
- Class 12 Maths Chapter 6 Notes
NCERT Solutions for Class 12 Maths - All Chapters
Below is the full chapter map for Collegedunia's Class 12 Maths NCERT Solutions library.
| Chapter | Solutions Link |
|---|---|
| Chapter 1 | Relations and Functions Solutions |
| Chapter 2 | Inverse Trigonometric Functions Solutions |
| Chapter 3 | Matrices Solutions |
| Chapter 4 | Determinants Solutions |
| Chapter 5 | Continuity and Differentiability Solutions |
| Chapter 7 | Integrals Solutions |
| Chapter 8 | Application of Integrals Solutions |
| Chapter 9 | Differential Equations Solutions |
| Chapter 10 | Vector Algebra Solutions |
| Chapter 11 | Three Dimensional Geometry Solutions |
| Chapter 12 | Linear Programming Solutions |
| Chapter 13 | Probability Solutions |
this chapter: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Application of Derivatives Chapter
The Application of Derivatives chapter splits into 3 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 6.1 | Rate of change of quantities |
| Exercise 6.2 | Increasing and decreasing functions |
| Exercise 6.3 | Maxima and minima |
| Miscellaneous Exercise | Mixed applications of derivatives |
PDF Download Formats and Languages for the Application of Derivatives Chapter
The Application of Derivatives Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The application of derivatives class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the application of derivatives class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The application of derivatives class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The application of derivatives class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The application of derivatives class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the application of derivatives class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Application of Derivatives Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Application of Derivatives chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable application of derivatives class 12 important questions you will see on board day.
- application of derivatives class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The application of derivatives class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching application of derivatives class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Application of Derivatives chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full application of derivatives class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Application of Derivatives Notes Pair with NCERT Solutions and the Formula Sheet
The Application of Derivatives Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Application of Derivatives Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| the PDF PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| application of derivatives class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The this chapter cover every back-of-chapter exercise plus the miscellaneous exercise.
- The application of derivatives class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The application of derivatives class 12 formulas reference sheet is the same A4 file students sometimes refer to as this Class 12 page all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 application of derivatives solutions, class 12 application of derivatives ncert solutions, ncert class 12 application of derivatives solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to the chapter notes |
|---|---|
| RD Sharma Class 12 Application of Derivatives | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Application of Derivatives | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the PDF | Free online walkthroughs; useful for video-style learning. |
| Shaalaa application of derivatives class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board this chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Application of Derivatives | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Application of Derivatives Notes Page Most Effectively
The recommended study plan for these notes chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use these notes PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The application of derivatives class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
All NCERT Solutions for Application of Derivatives Ex 6.2 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 6 Application of Derivatives Ex 6.2 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Show that the function given by f(x) = 3x + 17 is increasing on R.
Show that the function given by f(x) = e2x is increasing on R.
Show that the function given by f(x) = sin x is
(a) increasing in (0, π2)
(b) decreasing in (π2, π)
(c) neither increasing nor decreasing in (0, π).
Find the intervals in which the function f given by
f(x) = 2x2 - 3x is
(a) increasing (b) decreasing.
Find the intervals in which the function f given by
f(x) = 2x3 - 3x2 - 36x + 7 is
(a) increasing (b) decreasing.
Find the intervals in which the following functions are
strictly increasing or decreasing:
(a) x2 + 2x - 5
(b) 10 - 6x - 2x2
(c) -2x3 - 9x2 - 12x + 1
(d) 6 - 9x - x2
(e) (x+1)3(x-3)3.
Show that y = log(1+x) - 2x2+x, x > -1, is an increasing function of x throughout its domain.
Find the values of x for which y = [x(x-2)]2 is an increasing function.
Prove that y = 4 sin θ2 + cos θ - θ is an increasing function of θ in [0, π2].
Prove that the logarithmic function is increasing on (0, ∞).
Prove that the function f given by f(x) = x2 - x + 1 is neither strictly increasing nor decreasing on (-1, 1).
Which of the following functions are decreasing on (0,
π2)?
(A) cos x (B) cos 2x (C) cos 3x (D) tan x.
On which of the following intervals is the function f
given by f(x) = x100 + sin x - 1 decreasing?
(A) (0, 1) (B) (π2, π)
(C) (0, π2) (D) None of these.
For what values of a the function f given by f(x) = x2 + ax + 1 is increasing on [1, 2]?
Let I be any interval disjoint from [-1, 1]. Prove that the function f given by f(x) = x + 1x is increasing on I.
Prove that the function f given by f(x) = log sin x is increasing on (0, π2) and decreasing on (π2, π).
Prove that the function f given by f(x) = log|cos x| is decreasing on (0, π2) and increasing on (3π2, 2π).
Prove that the function given by f(x) = x3 - 3x2 + 3x - 100 is increasing in R.
The interval in which y = x2 e-x is increasing is
(A) (-∞, ∞) (B) (-2, 0) (C) (2, ∞)
(D) (0, 2).
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Application of Derivatives Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 6.2 of Class 12 Maths Chapter 6?
Ans. Exercise 6.2 contains 19 questions covering increasing and decreasing functions, with a mix of short-answer problems, long-answer derivations, and two MCQs at the end.
Ques. What is the main concept in Class 12 Maths Exercise 6.2?
Ans. The main concept is identifying intervals on which a function is increasing or decreasing using the sign of its first derivative, applied to polynomial, trigonometric, exponential and logarithmic functions.
Ques. How do I prove a function is strictly increasing on R?
Ans. Compute f'(x) and show that f'(x) > 0 for every real number x. If the strict inequality holds throughout the real line, the function is strictly increasing on R.
Ques. Are these NCERT Solutions for Exercise 6.2 aligned with the 2026-27 syllabus?
Ans. Yes. All Collegedunia solutions for Class 12 Maths Chapter 6 Exercise 6.2 follow the latest 2026-27 NCERT syllabus and CBSE board pattern.
Ques. Is Exercise 6.2 important for JEE Main?
Ans. Yes. Monotonicity and the first-derivative test appear in 2-3 JEE Main calculus questions every year, and the techniques drilled in Exercise 6.2 transfer directly.
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