When approached randomly, marks are lost in steps and structure. However, if you prepare strategically, Calculus becomes the most controllable scoring section of the paper. After all, the board consistently frames it around formulas, clear working, and application-based reasoning.
In this guide, our experts at Aakash break Calculus down in that exact context, aligned with weightage, recurring question patterns, and preparation strategies that actually help.
Quick Exam Facts: Calculus in CBSE Class 12 Maths
Before we break down formulas and scoring zones, you need clarity on the exam structure.
| Aspect | Details |
| Unit Calculus Weightage | 35/80 marks (44%) [It may vary by 5-10 marks depending on the pattern of the question paper.] |
| Chapters Included | 1. Continuity and Differentiability
2. Applications of Derivatives 3. Integrals 4. Applications of the Integrals 5. Differential Equations |
| Question Types | MCQs (20%), Short Answer (30%), Long Answer (50%) |
| Application-Based Weightage | Approximately 50% of the questions test the application |
| Pro Tip | Step marking matters. Even if the final answer is wrong, the correct method and formula fetch marks. |
Important Topics in Calculus (Strictly as per CBSE Syllabus)
See, focusing on what is actually included is very important. So, it’s strictly important to align your preparation to the prescribed CBSE Class 12 Maths syllabus.
| Unit III: Calculus | ||
| Chapter | Included | Excluded |
| 1. Continuity and Differentiability |
|
|
| 2. Applications of Derivatives |
|
|
| 3. Integrals |
|
|
| 4. Applications of Integrals |
|
|
| 5. Differential Equations |
|
|
Keeping your preparation strictly aligned with the CBSE Class 12 Maths syllabus is a non-negotiable. It will help you prevent unnecessary confusion and save some genuine revision time.
Chapter-Wise Scoring: Important Formulas
Each chapter in Calculus carries its own pattern of questioning. Here is a list of important formulas and structures that are often a make-or-break:
1. Continuity and Differentiability
| Concept | Formula |
| Continuity at x = a | f(a) exists; lim (x→a⁻) f(x) = lim (x→a⁺) f(x); lim (x→a) f(x) = f(a) |
| Product Rule | (uv)’ = u’v + uv’ |
| Quotient Rule | (u/v)’ = (v u’ − u v’)/v² |
| Chain Rule | If y = f(g(x)), then dy/dx = f’(g(x)) · g’(x) |
| Exponential | d/dx (eˣ) = eˣ ; d/dx (aˣ) = aˣ ln a |
| Logarithmic | d/dx (ln x) = 1/x |
| Inverse Trigonometric | d/dx (sin⁻¹x) = 1/√(1−x²); d/dx (cos⁻¹x) = −1/√(1−x²); d/dx (tan⁻¹x) = 1/(1+x²) |
| Logarithmic Differentiation | If y = [f(x)]^{g(x)}, take log on both sides and differentiate |
| Parametric Form | dy/dx = (dy/dt)/(dx/dt) |
| Second Derivative | d²y/dx² = d/dx (dy/dx) |
2. Applications of Derivatives
| Concept | Formula |
| Rate of Change | If y = f(x), then rate of change = dy/dx |
| Increasing Function | f’(x) > 0 on interval |
| Decreasing Function | f’(x) < 0 on interval |
| Critical Point | f’(x) = 0 or undefined |
| First Derivative Test | Sign change + to − → local maximum; − to + → local minimum |
| Second Derivative Test | If f’(a)=0 and f’’(a)>0 → minimum; f’’(a)<0 → maximum |
| Absolute Extrema | Compare values at critical points and interval endpoints |
3. Integrals
| Concept | Formula |
| Basic Integral | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1) |
| Exponential | ∫ eˣ dx = eˣ + C |
| Logarithmic | ∫ eˣ dx = eˣ + C |
| Trigonometric | ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C |
| Substitution | ∫ f(g(x))g’(x) dx = ∫ f(u) du |
| Integration by Parts | ∫ u dv = uv − ∫ v du |
| Partial Fractions | For rational functions P(x)/Q(x) |
| Definite Integral | ∫ₐᵇ f(x) dx = F(b) − F(a) |
| Properties | ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx |
4. Applications of Integrals
| Concept | Formula |
| Area under the curve (x-axis) | Area = ∫ₐᵇ f(x) dx |
| Area under curve (y-axis) | Area = ∫ₐᵇ f(y) dy |
| Standard Curves | Circle: x² + y² = a²;
Parabola: y² = 4ax or x² = 4ay; Ellipse: x²/a² + y²/b² = 1 |
5. Differential Equations
| Concept | Formula |
| Order | Highest order derivative present |
| Degree | Power of the highest order derivative |
| Variable Separable | dy/dx = g(x)h(y) → dy/h(y) = g(x) dx |
| Solution (Separable) | ∫ dy/h(y) = ∫ g(x) dx + C |
| Homogeneous Equation | dy/dx = F(y/x) |
| Substitution | y = vx → dy/dx = v + x dv/dx |
| Linear Differential Equation | dy/dx + P(x)y = Q(x) |
| Integrating Factor (IF) | IF = e^{∫P(x) dx} |
| General Solution | y·IF = ∫ Q(x)·IF dx + C |
Examples of Board-Level Questions from Calculus
Here are a few examples that reflect the type and structure commonly seen in board examinations from Calculus.
1. Find the value of k such that
f(x) = kx + 3, x ≤ 2
x^2 − 1, x > 2
is continuous at x = 2.
For continuity:
LHL = RHL
2k + 3 = 3
2k = 0
k = 0
2. Evaluate
Integral of x e^x dx
Using integration by parts:
Let
u = x → du = dx
dv = e^x dx → v = e^x
Integral of x e^x dx
= x e^x − Integral of e^x dx
= x e^x − e^x + C
= e^x (x − 1) + C
See, the point is if you want a stronger command over these chapters, regular practice is very important. You can use structured CBSE Maths questions with solutions for proper practice.
Tips for Preparing Calculus
Calculus is not about solving difficult questions. It is about avoiding avoidable mistakes. Here are some tips to keep in mind.
- Keep your integration formulas and differentiation rules revised regularly.
- In continuity questions, always show LHL, RHL and value at the point clearly.
- In maxima–minima, don’t stop at finding critical points, apply the test properly as well..
- Never forget “+ C” in indefinite integrals.
- Practice mixed questions from application of derivatives class 12 and differential equations class 12
- exercises instead of solving only one type repeatedly.
Before the exam, revise one consolidated sheet of derivative and integration formula expressions instead of flipping through the entire book.
Conclusion
Calculus accounts for a major share of marks in CBSE Class 12 Maths. The pattern is consistent, the methods are defined, and the marking is step-based. If formulas are clear and working is systematic, this unit can become your scoring point rather than another stressful topic.
FAQs
1. Is NCERT enough for Continuity and Differentiability?
Yes. Most board questions are directly aligned with NCERT examples and miscellaneous problems. Practising from continuity and differentiability class 12 ncert solutions is usually sufficient for board-level preparation.
2. How important is integration by parts in board exams?
Integration by parts, often referred to through the integration product rule, is frequently tested in 3–4 mark questions. It should be practiced in standard forms like ∫ x eˣ dx or ∫ x sin x dx.
3. Which part of Calculus is usually the most scoring?
Applications of derivatives and basic separable equations from differential equations class 12 are considered relatively structured and scoring when steps are written clearly.
