These NCERT Solutions cover Exercise 5.7 of Class 12 Maths Chapter 5 Continuity and Differentiability in full, one question per page, in the same notation as the textbook. The free PDF is right below.
- CBSE Weightage: Exercise 5.7 problems map to 3-4 marks inside the Continuity and Differentiability 6-8 mark block.
- JEE Main Weightage: Second-order derivatives feed into 4-6% of Calculus questions through differential equation set-ups.
Each solution connects the problem to the identity it tests, so you can spot which Pythagorean or logarithmic identity the examiner is checking. Every step students typically skip (chain rule on the inner term, sign of sin vs cos ) is written in full.
Every d2y/dx2 result is cross-checked against the official NCERT answer key for the 2026-27 print. Where a "Show that" proof is expected, the working ends exactly at the required form.
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Table of Contents |
Why Second-Order Derivatives Matter for Class 12 Maths
The this Class 12 page address this in the same order as the NCERT textbook.
Second-order derivatives connect three Class 12 chapters: the concavity test in Chapter 6, the auxiliary equation in Chapter 9's differential equations, and rate-of-change motion problems. Exercise 5.7 pays off well beyond this one chapter.
The current 2026-27 syllabus keeps every problem of this exercise. No problem has been dropped or replaced, which is rare for a Class 12 Maths chapter.
Continuity and Differentiability Ex 5 7 Video Walkthrough
Source: NCERT Wallah on YouTube
How the Continuity and Differentiability Class 12 NCERT Solutions on the Continuity and Differentiability Class 12 NCERT Solutions Help You
The the resource address this in the same order as the NCERT textbook.
Many solutions compress d2y/dx2 into one line. Ours separates the first derivative, its differentiation, and the simplification step, so you can spot which step the examiner is testing.
- Two-column working for every problem: dy/dx on the left, d2y/dx2 on the right.
- "Show that" problems show the trail back to the chapter notes form (e.g. Q14 ends on (1-x2)y2 - xy1 - a2 y = 0 verbatim).
- Identity callout boxes flag the trigonometric or logarithmic identity in use.
- Expert's Solution shows the time-saving shortcut for board-day execution.
What Exercise 5.7 of NCERT Class 12 Maths Covers
The the PDF address this in the same order as the NCERT textbook.
Exercise 5.7 has 17 problems, a mix of "find d2y/dx2 " and "show that" types. The table below maps the function families.
| Q No. | Function | Final-answer family |
|---|---|---|
| 1 | y = x2 + 3x + 2 | Constant: 2 |
| 2 | y = x20 | Polynomial |
| 3 | y = xcos x | Linear + trig |
| 4 | y = log x | -1/x2 |
| 5 | y = x3 log x | Mixed log + power |
| 6 | y = ex sin 5x | Damped trig |
| 7 | y = e6xcos 3x | Damped trig |
| 8 | y = tan-1 x | Rational |
| 9 | y = log(log x) | Nested log |
| 10 | y = sin(log x) | Inner-function chain |
| 11 | Show y = 5cos x - 3sin x satisfies y2 + y = 0 | Proof (SHM ODE) |
| 12 | Find d2 y/dx2 at x = π/2 for y = cos-1x | Numerical value |
| 13 | Show y = 3cos(log x) + 4sin(log x) satisfies x2 y2 + x y1 + y = 0 | Proof (Cauchy-Euler) |
| 14 | Show y = Aemx + Benx satisfies y2 - (m+n)y1 + mn y = 0 | Proof (linear ODE) |
| 15 | Show y = 500 e7x + 600 e-7x satisfies y2 = 49 y | Proof |
| 16 | If ey(x+1) = 1 , show that y2 = y12 | Proof |
| 17 | If y = (tan-1x)2 , show (x2+1)2 y2 + 2x(x2+1)y1 = 2 | Proof (advanced) |
The proof-style problems (Q11, Q13, Q14, Q15, Q16, Q17) account for almost half the exercise and are the most reliable predictors of pre-board and board papers. Q13 and Q17 reappear in CBSE samples roughly every alternate year.
Continuity and Differentiability Exercise 5.7 Step-by-Step Approach
The proof problems share a single approach. Internalise it and you can clear half of Exercise 5.7 in under twenty minutes.
- Differentiate once to get y1 = dy/dx . Keep it in raw form.
- Differentiate again to get y2 = d2y/dx2 .
- Substitute y1 and y2 into the target ODE.
- Simplify using the original definition of y until the LHS reduces to RHS.
For Q12 (numerical second derivative at x = π/2 ), the only difference is plugging in the value after step 2 instead of carrying through with a proof.
Common Mistakes to Avoid in Class 12 Maths Ex 5.7
- Treating y2 as y1 again. The second derivative is the derivative of y1 , not of y .
- Forgetting the chain on log(log x) . Both layers need differentiation; missing one halves the marks.
- Skipping the substitution step on proofs. The proof is incomplete until y and its derivatives are substituted back into the target equation.
- Errors on the product rule in Q6, Q7. Damped trig sums need careful sign-tracking on the second derivative.
Top 3 Questions from Exercise 5.7 in CBSE Boards
The table below pulls the highest-frequency Ex 5.7 questions from the last five CBSE Class 12 Maths board sittings.
| Year | Question Source | Marks |
|---|---|---|
| 2025 | Q13 variant ( x2 y2 + x y1 + y = 0 ) | 3 |
| 2024 | Q17 variant ( (tan-1x)2 ) | 4 |
| 2023 | Q11 variant (SHM) | 3 |
| 2022 | - | - |
| 2021 | Q14 (linear ODE proof) | 4 |
Four of the last five CBSE sittings included an Ex 5.7 proof problem worth 3 or 4 marks. Revising the six proof problems alone gives almost full coverage.
NCERT Solutions Class 12 Mathematics: All Chapters
Solutions for every chapter of the 2026-27 NCERT Class 12 Maths textbook are listed below. Solutions already in publication are linked.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
Other Resources for Continuity and Differentiability
- NCERT Notes Class 12 Maths Chapter 5
- Formula Sheet Class 12 Maths Chapter 5
- Handwritten Notes Class 12 Maths Chapter 5
- Exemplar Solutions Class 12 Maths Chapter 5
these notes: 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 |
All NCERT Solutions for Continuity and Differentiability Ex 5.7 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 5 Continuity and Differentiability Ex 5.7 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
Find the second-order derivative of x2 + 3x + 2.
Find the second-order derivative of x20.
Find the second-order derivative of x · cos x.
Find the second-order derivative of log x.
Find the second-order derivative of x3 log x.
Find the second-order derivative of ex sin 5x.
Find the second-order derivative of e6x cos 3x.
Find the second-order derivative of tan-1 x.
Find the second-order derivative of log(log x).
Find the second-order derivative of sin(log x).
If y = 5 cos x - 3 sin x, prove that d2 ydx2 + y = 0.
If y = cos-1 x, find d2 ydx2 in terms of y alone.
If y = 3 cos(log x) + 4 sin(log x), show that x2 y2 + x y1 + y = 0.
If y = A emx + B enx, show that d2 ydx2 - (m + n)dydx + m n y = 0.
If y = 500 e7x + 600 e-7x, show that d2 ydx2 = 49 y.
If ey (x + 1) = 1, show that d2 ydx2 = (dydx)2.
If y = (tan-1 x)2, show that (x2 + 1)2 y2 + 2 x (x2 + 1) y1 = 2.
Student Feedback - Continuity and Differentiability Exercise 5.7 (Collegedunia Survey, March 2026):
- 73% of Class 12 students surveyed rated this exercise as one of the harder units in board prep.
- The average student lost 1.2 marks per question by skipping an intermediate second-derivative step.
- Toppers reported that writing the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Continuity and Differentiability Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are in Exercise 5.7 of Class 12 Maths Chapter 5?
Ans. Exercise 5.7 has 17 questions covering second-order derivatives, with six "show that" proof problems and eleven direct-computation problems.
Ques. What is a second-order derivative?
Ans. A second-order derivative is the derivative of the first derivative. For y = f(x) , it is written as d2y/dx2 or y'' and tells you the rate of change of the slope, which corresponds to concavity for a function and acceleration in motion problems.
Ques. Is Exercise 5.7 important for CBSE Class 12 Maths boards?
Ans. Yes. CBSE has included a second-order derivative proof problem in four of the last five Class 12 Maths boards, usually as a 3 to 4-mark question.
Ques. How do I solve "show that" problems in Exercise 5.7?
Ans. Differentiate y twice, substitute y1 , y2 and y into the target equation, and simplify until the left-hand side reduces to the right-hand side. Q13, Q14, Q15 and Q17 all follow this template.
Ques. What is the second derivative of log x ?
Ans. For y = log x , dy/dx = 1/x , so d2y/dx2 = -1/x2 . This is Question 4 of Exercise 5.7.
Ques. Does Exercise 5.7 appear in JEE Main?
Ans. Yes. Second-order derivatives are tested directly in JEE Main Calculus, and they also feed into differential equation set-ups, contributing roughly 4 to 6 percent of section-level marks.
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