The Application of Derivatives Class 12 Formula Sheet page compiles NCERT Class 12 Mathematics Chapter 6 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.
- CBSE Weightage: Application of Derivatives carries 6-8 marks in CBSE Class 12 Maths boards.
- JEE Main Weightage: Application of Derivatives contributes 4-6% of Calculus questions in JEE Main 2026.
- JEE Main Weightage: Not part of the JEE Main syllabus; relevant only for CBSE and JEE aspirants.
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 formula sheet is laid out sub-topic by sub-topic in the order NCERT teaches them. Every formula carries a one-line "where used" tag so you can pull just the Application of Derivatives Class 12 Formulas for the question pattern in front of you.
Collegedunia's editors have cross-verified each entry against the official NCERT 2026-27 print and the JEE Main Calculus syllabus. Where a formula has multiple standard forms (e.g. tangent equation in slope-point form vs slope-intercept form), the sheet lists both.
Application of Derivatives Chapter Marks Distribution Across Exams
These notes address this in the same order as the NCERT textbook.
The chapter's weight is consistent across the three exam streams. Below is the year-wise marks pattern in CBSE Class 12 Maths since 2021.
| Year | CBSE Class 12 Maths marks | JEE Main calculus share |
|---|---|---|
| 2025 | 5 marks (optimisation) | 2 questions (~5%) |
| 2024 | 3 marks (rate of change) | 2 questions (~5%) |
| 2023 | 5 marks (local extrema) | 3 questions (~6%) |
| 2022 | 4 marks (tangent-normal) | 2 questions (~4%) |
| 2021 | 3 marks (increasing-decreasing) | 2 questions (~5%) |
The five-year average is 4 marks in CBSE and 2.2 questions in JEE Main. Application of Derivatives is one of the most reliably tested chapters in Class 12 Mathematics.
Application of Derivatives Video Walkthrough
Source: Magnet Brains on YouTube
How the Application of Derivatives Class 12 Formulas on the Application of Derivatives Class 12 Formulas Help You
The Collegedunia formula sheet is a single-pass revision tool. Three design choices compress the this Class 12 page into 7 pages.
- Formula + use-case pairing: each formula carries a "where used" tag, so you do not need a separate revision book to know which formula to use when.
- Common-variable callouts: every formula explicitly names the variable (e.g. r for radius, h for height) so substitution errors drop.
- Two-form formulae listed: where multiple forms exist (tangent equation, normal equation), both are shown to match whatever the question uses.
- Geometric formulae included: volume of sphere, cone, cube and surface area formulae are listed in one panel for rate-of-change and optimisation problems.
Class 12 Application of Derivatives All Formulas in One Place
Every formula needed for Chapter 6 is consolidated below in five sub-topic groups. This is the canonical formula reference for the the resource.
Group 1: Rate of Change
| Concept | Formula | Where used |
|---|---|---|
| Instantaneous rate of change | dydx at x = x0 | Rate problems |
| Related rates link | dydt = dydx · dxdt | Two-quantity rate problems |
| Average rate of change | f(b) - f(a)b - a | Comparison with instantaneous rate |
Group 2: Increasing and Decreasing Functions
| Concept | Formula / Condition | Where used |
|---|---|---|
| Strictly increasing on (a, b) | f'(x) > 0 for all x ∈ (a, b) | Interval determination |
| Strictly decreasing on (a, b) | f'(x) < 0 for all x ∈ (a, b) | Interval determination |
| Monotone (non-strict) | f'(x) ≥ 0 or f'(x) ≤ 0 | Proof-type questions |
Group 3: Tangents and Normals
| Concept | Formula | Where used |
|---|---|---|
| Slope of tangent at (x0, y0) | m = f'(x0) | Tangent equation |
| Equation of tangent | y - y0 = m(x - x0) | Line through curve point |
| Slope of normal | -1m (provided m ≠ 0 ) | Normal equation |
| Equation of normal | y - y0 = -1m(x - x0) | Line perpendicular to tangent |
| Tangent parallel to x-axis | f'(x0) = 0 | Special-direction tangent |
| Tangent parallel to y-axis | f'(x0) undefined | Special-direction tangent |
Group 4: Maxima and Minima
| Concept | Formula / Test | Where used |
|---|---|---|
| Critical point | f'(c) = 0 or f'(c) undefined | Candidate extrema |
| First derivative test (max) | f'(x) : + to - at c | Local maximum |
| First derivative test (min) | f'(x) : - to + at c | Local minimum |
| Second derivative test (max) | f'(c) = 0, f''(c) < 0 | Local maximum (quick) |
| Second derivative test (min) | f'(c) = 0, f''(c) > 0 | Local minimum (quick) |
| Absolute extremum on [a, b] | Compare f at all critical pts and at endpoints a, b | Closed-interval optimisation |
Group 5: Differentials and Approximations
| Concept | Formula | Where used |
|---|---|---|
| Differential | dy = f'(x) dx | Small-change estimate |
| Linear approximation | Δ y ≈ f'(x) Δ x | Numerical approximation |
Group 6: Geometric Formulae (for word problems)
| Shape | Volume formula | Surface area formula |
|---|---|---|
| Sphere | V = 43π r3 | S = 4π r2 |
| Cone | V = 13π r2 h | S = π r l + π r2 (with slant l) |
| Cylinder | V = π r2 h | S = 2π r h + 2π r2 |
| Cube | V = a3 | S = 6a2 |
The geometric formulae are non-negotiable for 5-mark optimisation problems. The set above covers every standard CBSE word problem since 2018.
Top 5 Most-Used Formulae in CBSE Class 12 Application of Derivatives
The chapter notes address this in the same order as the NCERT textbook.
Of the 22 formulae in the sheet, five are responsible for almost all the marks awarded in CBSE Class 12 Maths boards since 2020.
| Rank | Formula | Used in |
|---|---|---|
| 1 | Second derivative test: f''(c) < 0 max, f''(c) > 0 min | 5-mark optimisation |
| 2 | Slope of tangent: m = f'(x0) | 3-4 mark tangent-normal |
| 3 | Related rates: dydt = f'(x) dxdt | 3-mark rate problems |
| 4 | Sphere volume: V = 43π r3 | Sphere rate problem |
| 5 | Strictly increasing/decreasing condition | 3-mark interval problem |
Memorising these five formulae alone covers an estimated 70-75% of the chapter's CBSE marks since 2020.
Class 12 Maths Application of Derivatives Quick-Recall Summary
The the PDF address this in the same order as the NCERT textbook.
30-second exam recap:
- Rate of change: dydx at the given point.
- Two related quantities: chain through time.
- Tangent slope = f'(x0) ; normal slope = -1/f'(x0) .
- Critical point: f'(c) = 0 . Classify with first or second derivative test.
- Closed interval: also test endpoints for absolute max/min.
Common Mistakes Class 12 Students Make with Application of Derivatives Formulae
The this chapter are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Wrong sign on normal slope. Normal slope is -1/m , not 1/m . A common 2-mark mistake.
- Missing the closed-interval endpoints. Without evaluating f at a and b, you cannot claim "absolute maximum".
- Confusing first and second derivative test conditions. Second derivative test: f''(c) < 0 is max, NOT min. Memorise.
- Forgetting the constraint in optimisation. Always reduce to one variable before differentiating.
Related Resources for Application of Derivatives
- NCERT Notes Class 12 Maths Chapter 6
- NCERT Solutions Class 12 Maths Chapter 6
- Application of Derivatives Handwritten Notes
- Exemplar Solutions Application of Derivatives
NCERT Formula Sheet Class 12 Mathematics: All Chapters
Formula sheets for every chapter of the 2026-27 NCERT Class 12 Maths textbook are listed below. Self-row Chapter 6 is excluded; published chapters are linked.
| Chapter | NCERT Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
these notes: 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 the resource important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching the chapter notes 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 the PDF 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 this chapter 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 |
| application of derivatives class 12 ncert solutions 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 application of derivatives class 12 ncert solutions 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 these notes all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board this Class 12 page 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 resource |
|---|---|
| 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 chapter notes | 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 the PDF 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 the this chapter 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 the application of derivatives class 12 ncert solutions 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 these notes 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.
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 Formulas - Frequently Asked Questions
Ques. How many formulae are in the Application of Derivatives formula sheet?
Ans. The Application of Derivatives formula sheet has 22 essential formulae grouped into 5 sub-topics: rate of change, increasing-decreasing functions, tangents and normals, maxima-minima, and approximations.
Ques. What is the formula for the slope of a tangent in Class 12 Maths?
Ans. The slope of the tangent to a curve y = f(x) at x = x0 is m = f'(x0) , the value of the first derivative at that point. The corresponding tangent line is y - y0 = m(x - x0) .
Ques. What is the second derivative test in Class 12 Maths?
Ans. At a critical point c where f'(c) = 0 : if f''(c) < 0 , c is a local maximum; if f''(c) > 0 , c is a local minimum; if f''(c) = 0 , the test is inconclusive and you fall back to the first derivative test.
Ques. Which formula is most asked from Application of Derivatives in CBSE?
Ans. The second derivative test is the highest-frequency formula in CBSE Class 12 Maths boards, since every 5-mark optimisation problem ends with classifying the critical point as max or min.
Ques. Do I need geometric formulae like sphere volume for Application of Derivatives?
Ans. Yes. CBSE word problems regularly use volume and surface area of sphere, cone, cylinder and cube. The formula sheet lists all four in a single panel so you can reach for the right one without flipping back to Class 9 or 10 notes.
Ques. What is the slope of the normal to a curve?
Ans. The slope of the normal at a point is the negative reciprocal of the tangent's slope, that is -1/f'(x0) , provided f'(x0) is non-zero. If the tangent is horizontal, the normal is vertical and has undefined slope.
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