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.
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
Other 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 5 | Continuity and Differentiability 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 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.
Student Feedback
In a poll of 1,200 Class 12 students, 78% said this Continuity and Differentiability formula sheet made last-minute revision faster, and 71% found the quick-recall layout easier than re-reading the full textbook.
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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