The Continuity and Differentiability Class 12 Formulas given here cover Class 12 Mathematics Chapter 5 Continuity and Differentiability in a compact, recall-friendly format. The chapter notes are split across the standard NCERT sections of limits, continuity at a point, differentiability and the chain rule, with each formula in the formula sheet presented with its assumptions explicitly stated.
- CBSE Weightage: 8 to 10 marks (typically one 5-mark long answer plus one 3-mark short answer or assertion-reason item).
- JEE Main Weightage: 4 to 6% of the Maths section (2 to 3 questions per paper).
- CUET (UG) Weightage: 2 to 3 MCQs in nearly every shift.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
Standard Derivatives learn Table for Class 12 Maths Chapter 5
The the PDF address this in the same order as the NCERT textbook.
The table below tabulates every standard derivative referenced in Class 12 Maths Chapter 5, mapped to the NCERT section where it is introduced. Every row has surfaced in a CBSE or JEE Main question in the last five years.
| Function (f(x) | Derivative f'(x) | NCERT Section | Domain Note |
|---|---|---|---|
| c (constant) | 0 | 5.2 | All x |
| xn | n xn-1 | 5.2 | n ∈ R |
| √x | 12√x | 5.2 | x > 0 |
| 1x | -1x2 | 5.2 | x ≠ 0 |
| sin x | cos x | 5.3 | All x |
| cos x | -sin x | 5.3 | All x |
| tan x | sec2 x | 5.3 | x ≠ (2k+1)π2 |
| cot x | -csc2 x | 5.3 | x ≠ kπ |
| sec x | sec x tan x | 5.3 | x ≠ (2k+1)π2 |
| csc x | -csc x cot x | 5.3 | x ≠ kπ |
| sin-1 x | 1√1 - x2 | 5.3 | |x| < 1 |
| cos-1 x | -1√1 - x2 | 5.3 | |x| < 1 |
| tan-1 x | 11 + x2 | 5.3 | All x |
| cot-1 x | -11 + x2 | 5.3 | All x |
| sec-1 x | 1|x|√x2 - 1 | 5.3 | |x| > 1 |
| csc-1 x | -1|x|√x2 - 1 | 5.3 | |x| > 1 |
| ex | ex | 5.4 | All x |
| ax | ax ln a | 5.4 | a > 0 |
| ln x | 1x | 5.4 | x > 0 |
| a x | 1x ln a | 5.4 | x > 0, a > 0, a ≠ 1 |
| |x| | x|x| (i.e. sign of x) | 5.2 (extension) | x ≠ 0 |
Memorising these 21 standard derivatives covers roughly 70% of the marks budget for the this chapter across the last five CBSE and JEE Main papers.
Continuity and Differentiability Video Walkthrough
Source: NCERT Wallah on YouTube
Differentiation Rules: Sum, Product, Quotient and Chain Rule Formulas
These notes address this in the same order as the NCERT textbook.
The rules below combine the standard derivatives above into any closed-form function. CBSE long-answer questions almost always test the chain rule layered with one of the other three.
| Rule | Formula | Typical Application |
|---|---|---|
| Sum / Difference rule | (u ± v)' = u' ± v' | Term-by-term differentiation |
| Scalar multiple | (k · u)' = k · u' | Constant factor pull-out |
| Product rule | (uv)' = u'v + uv' | x sin x , ex cos x |
| Quotient rule | (uv)' = u'v - uv'v2 | sin xx , rational functions |
| Chain rule (single layer) | ddx[f(g(x))] = f'(g(x)) · g'(x) | sin(x2) , etan x |
| Chain rule (two layers) | ddx[f(g(h(x)))] = f'(g(h(x))) · g'(h(x)) · h'(x) | sin(√cos x) , JEE Main level |
| Power-chain composite | ddx[g(x)]n = n [g(x)]n-1 · g'(x) | (1 + x2)5 , √1 + sin x |
| Exponential-chain composite | ddx eg(x) = eg(x) · g'(x) | esin x , ex2 |
| Logarithmic-chain composite | ddx ln g(x) = g'(x)g(x) | ln(sec x + tan x) |
Continuity and Differentiability Quick-Recall Formula Cards
The this Class 12 page address this in the same order as the NCERT textbook.
The cards below collapse the learn tables into the six highest-use formulae from Chapter 5. Memorise them as raw expressions; CBSE 1-mark MCQs test one of these almost every year.
How the Continuity and Differentiability Class 12 Formulas on the Continuity and Differentiability Class 12 Formulas Help You
This sheet is built around the five habits CBSE toppers share when revising Chapter 5: standard-derivative recall, chain-rule layering, the parametric and implicit templates, logarithmic differentiation, and the three-step continuity check.
- 2026-27 NCERT Alignment: Every derivative matches the current edition; Rolle's Theorem and the Mean Value Theorem appear in sections 5.7 and 5.8 with full statements.
- Inverse Trigonometric Block on One Page: All six inverse trig derivatives are stacked with their domain restrictions, so the sign and the |x| prefix never get confused under time pressure.
- Chain-Rule Decision Tree: Single-layer, two-layer, power, exponential and logarithmic composites are listed as separate rows so you can match the question shape to the right template in seconds.
- Logarithmic Differentiation Steps: The three-step protocol for y = f(x)g(x) sits in its own block, in the exact sequence CBSE awards the 5 marks.
Continuity at a Point: The Three-Step Formula Check
A function f is continuous at x = a if and only if all three conditions below hold. Missing any one of them, in the order CBSE expects, costs at least one mark of a 5-mark question.
| Step | Condition | What It Tests |
|---|---|---|
| 1 | f(a) is defined | Existence of the function value at the point |
| 2 | x → a- f(x) = x → a+ f(x) = L | Left-hand limit equals right-hand limit |
| 3 | L = f(a) | The common limit equals the function value |
For a piecewise function with parameters, write the three steps as separate sub-bullets. CBSE marking schemes award one mark per step and one for the final conclusion.
Logarithmic Differentiation Template for y = f(x)g(x) Functions
Whenever the function has a variable in both base and exponent (the classic CBSE 5-mark question), logarithmic differentiation is the only safe route. The three-step template below mirrors the marking scheme.
- Step 1, Take logarithms (1 mark): Apply ln to both sides. ln y = g(x) · ln f(x) . Use only the natural log; mixing bases costs marks.
- Step 2, Differentiate implicitly (2 marks): Use the product rule on the right-hand side and 1y · dydx on the left. 1ydydx = g'(x) ln f(x) + g(x) · f'(x)f(x) .
- Step 3, Solve for dydx (2 marks): Multiply both sides by y and re-substitute y = f(x)g(x). dydx = f(x)g(x) [ g'(x) ln f(x) + g(x) f'(x)f(x) ] .
This 5-mark question has appeared in CBSE Class 12 Maths in every paper from 2021 to 2025, usually as y = xx , y = (sin x)x , or y = xsin x .
Parametric and Implicit Differentiation Formula Block
Two further question types recur every year: parametric form (x and y as functions of t) and implicit form (x and y inside one equation). Both use the chain rule, but the bookkeeping differs.
| Type | Formula | Sample Question |
|---|---|---|
| Parametric (first derivative) | dydx = dy/dtdx/dt , provided dx/dt ≠ 0 | x = acos t, y = asin t |
| Parametric (second derivative) | d2ydx2 = ddt(dydx) · 1dx/dt | x = at2, y = 2at |
| Implicit (single equation) | Differentiate both sides w.r.t. x, treat y as y(x), then collect dydx | x2 + y2 = a2 |
| Implicit (with product term) | Use the product rule on xy terms: ddx(xy) = y + xdydx | x2 + 2xy + y3 = 7 |
Rolle's Theorem and Mean Value Theorem: Formulas and Hypotheses
The two theorems below close the the resource and carry a recurring 2 to 3 mark CBSE question.
| Theorem | Hypotheses | Conclusion |
|---|---|---|
| Rolle's Theorem | f continuous on [a, b], differentiable on (a, b), and f(a) = f(b) | ∃ c ∈ (a, b) with f'(c) = 0 |
| Mean Value Theorem (Lagrange) | f continuous on [a, b], differentiable on (a, b) | ∃ c ∈ (a, b) with f'(c) = f(b) - f(a)b - a |
The CBSE board paper most often asks you to verify the hypotheses, then locate c by solving f'(c) = 0 (Rolle) or f'(c) = f(b) - f(a)b - a (MVT). Always state each hypothesis check on a separate line.
Top 5 Most-Asked Topics from Continuity and Differentiability in CBSE Class 12 Board Exams
The ranking below shows which sub-topics of Chapter 5 have driven the most marks in CBSE Class 12 Maths papers from 2025 back to 2021.
- Logarithmic differentiation of f(x)g(x) (5 marks, every year from 2021 to 2025).
- Continuity of a piecewise function with a parameter to be determined (5 marks, 4 of last 5 years).
- Second derivative via parametric form (5 marks, 3 of last 5 years).
- Chain rule on composite inverse trigonometric expressions (3 to 4 marks, recurring).
- Verification of Rolle's Theorem or MVT on a polynomial (2 to 3 marks, recurring).
Full year-wise PYQ map: Continuity and Differentiability Class 12 Maths NCERT Solutions
Memory Mnemonics for Class 12 Maths Continuity and Differentiability
Use these short-form mnemonics in the last hour before the exam. They turn abstract identities into concrete actions.
One-Shot Revision Tips for Class 12th Maths Chapter 5 Continuity and Differentiability
The five tips below summarise what to revise in the final 60 minutes before the board exam. Each one targets a recurring CBSE error pattern that costs Class 12 candidates marks every year.
- Write down each chain-rule layer separately: for sin(√cos x), list f = sin u, u = √v, v = cos x, then differentiate from the outside in.
- Quote the domain on every inverse-trig derivative: ddxsin-1 x = 1√1 - x2 is valid only for |x| < 1. CBSE marking schemes reserve a mark for the explicit domain.
- Always state continuity in three steps: existence of f(a), equal left and right limits, and equality with f(a). One mark per step.
- For f(x)g(x), take logs immediately: never apply the power rule with a variable exponent or the exponential rule with a variable base.
- Verify Rolle / MVT hypotheses on separate lines: continuity on [a, b], differentiability on (a, b), and (for Rolle) f(a) = f(b).
More Class 12 Maths Continuity and Differentiability Resources from Collegedunia
NCERT Formula Sheet for Class 12 Maths: All Chapters
Use the table below to jump to the Formula Sheet of any other Class 12 Maths chapter. Each link opens the same learn-table format you have on the chapter notes.
| Chapter | Resource |
|---|---|
| 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 6 | Application of Derivatives 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 |
the PDF: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Continuity and Differentiability Chapter
The Continuity and Differentiability chapter splits into 7 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 5.1 | Continuity at a point and on an interval |
| Exercise 5.2 | Algebra of continuous functions |
| Exercise 5.3 | Differentiability and chain rule |
| Exercise 5.4 | Derivatives of inverse trigonometric functions |
| Exercise 5.5 | Logarithmic differentiation |
| Exercise 5.6 | Parametric and implicit differentiation |
| Exercise 5.7 | Second-order derivatives; Rolle's and Mean Value Theorem |
| Miscellaneous Exercise | Mixed continuity and differentiability problems |
PDF Download Formats and Languages for the Continuity and Differentiability Chapter
The Continuity and Differentiability 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 continuity and differentiability 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 continuity and differentiability class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The continuity and differentiability class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The continuity and differentiability class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The continuity and differentiability 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 continuity and differentiability 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 Continuity and Differentiability Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Continuity and Differentiability 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 the chapter notes important questions you will see on board day.
- continuity and differentiability 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 PDF important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching continuity and differentiability 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 Continuity and Differentiability 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 this chapter 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 Continuity and Differentiability Notes Pair with NCERT Solutions and the Formula Sheet
These notes 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 |
|---|---|---|
| Continuity and Differentiability Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| continuity and differentiability class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| continuity and differentiability 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 continuity and differentiability class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The continuity and differentiability 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 continuity and differentiability class 12 formulas reference sheet is the same A4 file students sometimes refer to as continuity and differentiability class 12 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 continuity and differentiability solutions, class 12 continuity and differentiability ncert solutions, ncert class 12 continuity and differentiability 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 Continuity and Differentiability | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Continuity and Differentiability | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the chapter notes | Free online walkthroughs; useful for video-style learning. |
| Shaalaa continuity and differentiability 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 Continuity and Differentiability | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Continuity and Differentiability 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 continuity and differentiability 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.
Continuity and Differentiability Class 12 Formulas - Frequently Asked Questions
Ques. Where can I download the this chapter for free?
Ans. You can download the Class 12 Maths Chapter 5 Continuity and Differentiability Formula Sheet PDF directly from the download card on this page. Both the Normal and HD versions are free and cover every standard derivative, chain-rule template, logarithmic differentiation step, parametric and implicit formula, and the Rolle and Mean Value Theorem statements.
Ques. What are the standard derivatives I must memorise for Class 12 Maths Chapter 5?
Ans. The 21 standard derivatives in the learn table at the top of this page cover roughly 70% of these notes's mark budget. Priority order: the six trigonometric derivatives, the six inverse trigonometric derivatives, ex, ax, ln x, a x, the power rule xn, and √x. Memorise these as raw facts.
Ques. What is the chain rule formula in Class 12 Maths Chapter 5?
Ans. For a composite y = f(g(x)), the chain rule states dydx = f'(g(x)) · g'(x). For two layers y = f(g(h(x))), it extends to dydx = f'(g(h(x))) · g'(h(x)) · h'(x). Always list each layer separately before differentiating, especially for JEE Main two-layer compositions.
Ques. How do I differentiate functions of the form y = f(x)g(x)?
Ans. Use logarithmic differentiation. Step 1: take ln of both sides to get ln y = g(x) ln f(x). Step 2: differentiate implicitly, using the product rule on the right-hand side, to get 1ydydx = g'(x) ln f(x) + g(x) f'(x)f(x). Step 3: multiply both sides by y and substitute back y = f(x)g(x). This is the recurring 5-mark CBSE question.
Ques. What is the formula for parametric differentiation in Class 12 Maths?
Ans. If x = x(t) and y = y(t), then dydx = dy/dtdx/dt, provided dx/dt ≠ 0. For the second derivative, use d2ydx2 = ddt(dydx) · 1dx/dt. Do NOT use d2y/dt2d2x/dt2, this shortcut is wrong and is the single most common 5-mark CBSE error.
Ques. Is the Continuity and Differentiability formula sheet aligned with the 2026-27 NCERT?
Ans. Yes. Every formula matches the current 2026-27 syllabus for Class 12 Maths Chapter 5. Continuity, differentiability, the algebra of derivatives, chain rule, derivatives of implicit, inverse-trigonometric, exponential, logarithmic, and parametric functions, along with Rolle's Theorem and the Mean Value Theorem, are all retained in the new edition.
Ques. State Rolle's Theorem and the Mean Value Theorem for Class 12 Maths.
Ans. Rolle's Theorem: if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) with f'(c) = 0. Mean Value Theorem (Lagrange):
if f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) with f'(c) = f(b) - f(a)b - a. Verify each hypothesis on a separate line in the answer.
Ques. How do I prove that a piecewise function is continuous at a given point?
Ans. Use the three-step CBSE check. Step 1: confirm f(a) is defined. Step 2: compute the left-hand limit x → a- f(x) and the right-hand limit x → a+ f(x) separately and verify they are equal. Step 3: verify that the common limit equals f(a). For a parameter question, equate the three values and solve for the parameter. Each step carries one mark.
Ques. How many pages is the Class 12 Maths Chapter 5 Continuity and Differentiability Formula Sheet PDF?
Ans. The Continuity and Differentiability Formula Sheet PDF runs 9 pages and packs all 21 standard derivatives, the four algebra-of-derivatives rules, the seven chain-rule templates, the three-step logarithmic differentiation protocol, the parametric and implicit formula block, and full statements of Rolle's Theorem and the Mean Value Theorem, plus the quick-recall card grid.
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