Class 12 · Chapter 5 · Calculus unit (35 of 80 marks)

Continuity and Differentiability Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 73 board-style questions.

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73 questions
20 multiple choice, 3 assertion–reason, 20 very short answer, 20 short answer, 6 long answer, 4 case study
About 20 hours to master

Calculus unit: 35 of 80 theory marks (Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations).

NCERT textbook: Chapter 5 on ncert.nic.in (PDF)

Revision notes

Continuity and Differentiability — revision notes

1. Continuity

$f$ is continuous at $x = c$ (a point of its domain) if $lim_{x \to c^{-}} f (x) = lim_{x \to c^{+}} f (x) = f (c)$ . A function is continuous if it is continuous at every point of its domain; points outside the domain are not called discontinuities.

Polynomials, $\sin x, \cos x$ , $e^{x}$ , $\log x$ $(x > 0)$ , $| x |$ , rational functions (where defined) are continuous.
Sums, differences, products, quotients (denominator $\neq 0$ ) and composites of continuous functions are continuous.
Useful limits: $lim_{x \to 0} \frac{\sin x}{x} = 1$ , $lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2}$ .
$[x]$ (greatest integer) is discontinuous exactly at the integers.

2. Differentiability

$f^{'} (c) = lim_{h \to 0} \frac{f (c + h) - f (c)}{h}$ must exist, i.e. LHD $=$ RHD (finite). Differentiable $\Rightarrow$ continuous; the converse is false (e.g. $| x |$ at $0$ ). For a piecewise function, first check continuity, then compare the one-sided derivatives.

3. Rules

Chain rule: $\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)$ .
$\frac{d}{d x} \sin^{- 1} x = \frac{1}{\sqrt{1 - x^{2}}}$ , $\frac{d}{d x} \cos^{- 1} x = - \frac{1}{\sqrt{1 - x^{2}}}$ $(| x | < 1)$ , $\frac{d}{d x} \tan^{- 1} x = \frac{1}{1 + x^{2}}$ .
$\frac{d}{d x} e^{x} = e^{x}$ , $\frac{d}{d x} a^{x} = a^{x} \log a$ , $\frac{d}{d x} \log | x | = \frac{1}{x}$ .
Implicit: differentiate every term w.r.t. $x$ , treating $y$ as a function of $x$ ( $\frac{d}{d x} y^{2} = 2 y \frac{d y}{d x}$ ).
Logarithmic differentiation for $[u (x)]^{v (x)}$ : $\log y = v \log u$ . For a sum of such terms, differentiate each term separately.
Parametric: $\frac{d y}{d x} = \frac{d y / d t}{d x / d t}$ ; $\frac{d^{2} y}{d x^{2}} = \frac{\frac{d}{d t} (\frac{d y}{d x})}{d x / d t}$ (not $\frac{d^{2} y / d t^{2}}{d^{2} x / d t^{2}}$ ).

4. Inverse-trig simplification

Substitute $x = \sin θ, \cos θ$ or $\tan θ$ , simplify, and check that the resulting angle lies in the principal range before writing $\sin^{- 1} (\sin α) = α$ . If it does not, adjust (e.g. $π - α$ ); the derivative can change sign across such points.

Worked example 1

Is $f (x) = {\begin{cases} 3 x - 2, & x \leq 2 \\ x^{2}, & x > 2 \end{cases}$ differentiable at $2$ ? $f (2) = 4$ , right limit $4$ : continuous. LHD $= 3$ , RHD $= 4$ : not differentiable.

Worked example 2

$y = (\cos x)^{\sin x}$ , $0 < x < \frac{π}{2}$ : $\log y = \sin x \log \cos x \Rightarrow \frac{d y}{d x} = (\cos x)^{\sin x} [\cos x \log \cos x - \frac{\sin^{2} x}{\cos x}]$ .

Worked example 3

$x = a t^{2}, y = a t^{3}$ : $\frac{d y}{d x} = \frac{3 a t^{2}}{2 a t} = \frac{3 t}{2}$ ; $\frac{d^{2} y}{d x^{2}} = \frac{3 / 2}{2 a t} = \frac{3}{4 a t}$ .

Common errors

Declaring a function differentiable because the two formulas "join up" — continuity is not enough.
Using $\log (u + v) = \log u + \log v$ (false). Take logs only of products, quotients and powers.
Treating $x^{x}$ like $x^{n}$ or $a^{x}$ : neither rule applies; use logarithms.
Forgetting $\frac{d y}{d x}$ on $y$ -terms in implicit differentiation, or the inner derivative in the chain rule.
Dividing $\frac{d^{2} y}{d t^{2}}$ by $\frac{d^{2} x}{d t^{2}}$ for a parametric second derivative.

Board tips

In "find $k$ " continuity questions, write LHL, RHL and $f (c)$ explicitly — each carries marks.
In "prove that" second-order questions, differentiate the simplest form (e.g. multiply out $y_{1} \sqrt{1 - x^{2}} = \dots$ ) before differentiating again.
State the interval restriction whenever you use an inverse-trig substitution.
Rolle’s theorem and the Mean Value Theorem are no longer in the syllabus — do not spend time on them.

Topics in this chapter: Continuity · Differentiability · Chain rule and composite functions · Derivatives of inverse trigonometric functions · Exponential and logarithmic functions · Parametric differentiation · Second order derivatives · Logarithmic differentiation · Implicit differentiation.

Formulas for this chapter: Chapter 5 on the Class 12 formula sheet (free PDF and A3 poster).

Continuity and Differentiability Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 241 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceContinuity

For which value of $k$ is $f (x) = k x - 3$ (for $x \leq 4$ ), $f (x) = x + 5$ (for $x > 4$ ) continuous at $x = 4$ ?

Show worked answer

$f (4) = 4 k - 3$ and the right-hand limit is $4 + 5 = 9$ : $4 k - 3 = 9 \Rightarrow k = 3$ A1
Correct option: (a) $3$

Q2·1 mark·Multiple choiceDifferentiability and derivatives

The derivative of $\sin (\log x)$ , for $x > 0$ , is

Q3·1 mark·Assertion–reasonDifferentiability and derivatives

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): $f (x) = | x - 3 |$ is continuous at $x = 3$ but not differentiable there.

Reason (R): Every continuous function is differentiable.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerImplicit and parametric differentiation

The curve $x^{2} - x y + 2 y^{2} = 8$ defines $y$ implicitly as a function of $x$ . Find $\frac{d y}{d x}$ .

Show worked answer

Answer:

\frac{y - 2 x}{4 y - x}

Differentiate: $2 x - y - x \frac{d y}{d x} + 4 y \frac{d y}{d x} = 0$ M1
$\frac{d y}{d x} = \frac{y - 2 x}{4 y - x}$ A1

Q5·2 marks·Short answerDifferentiability and derivatives

Use the product rule to find the derivative of $e^{x^{2}} \cos x$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerImplicit and parametric differentiation

If $x = 2 t^{2}$ and $y = t^{3}$ , find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ in terms of $t$ , and their values at $t = 2$ .

Show worked answer

Answer:

\frac{3 t}{4}

\frac{3}{16 t}

;

\frac{3}{2}

and

\frac{3}{32}

$\frac{d x}{d t} = 4 t$ , $\frac{d y}{d t} = 3 t^{2}$ , so $\frac{d y}{d x} = \frac{3 t}{4}$ ( $= \frac{3}{2}$ at $t = 2$ ) M1
$\begin{aligned} \frac{d^{2} y}{d x^{2}} & = \frac{d}{d t} (\frac{3 t}{4}) \div \frac{d x}{d t} \\ = \frac{3}{4} \times \frac{1}{4 t} \\ = \frac{3}{16 t} \end{aligned}$ A1
At $t = 2$ : $\frac{3}{32}$ A1

Q7·3 marks·Short answerLogarithmic differentiation

Find $\frac{d y}{d x}$ if $y = (x^{2} + 1)^{x}$ .

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerSecond-order derivatives

Given $y = e^{3 x} (a \cos 2 x + b \sin 2 x)$ for constants $a$ and $b$ , prove that $y^{″} - 6 y^{'} + 13 y = 0$ .

Show worked answer

Answer: Shown

$y^{'} = 3 e^{3 x} (a \cos 2 x + b \sin 2 x) + e^{3 x} (- 2 a \sin 2 x + 2 b \cos 2 x)$ M1
$= 3 y + 2 e^{3 x} (b \cos 2 x - a \sin 2 x)$ A1
$y^{″} = 3 y^{'} + 6 e^{3 x} (b \cos 2 x - a \sin 2 x) + 2 e^{3 x} (- 2 b \sin 2 x - 2 a \cos 2 x)$ M1
$= 3 y^{'} + 3 (y^{'} - 3 y) - 4 y = 6 y^{'} - 13 y$ A1
So $y^{″} - 6 y^{'} + 13 y = 0$ A1

Q9·5 marks·Long answerContinuity

$f (x) = {\begin{cases} 3 x + a, & x < 1 \\ x^{2} + b x, & 1 \leq x \leq 3 \\ 12, & x > 3 \end{cases}$ is continuous for all real $x$ . (a) Find $a$ and $b$ . (b) Is $f$ differentiable at $x = 1$ ? At $x = 3$ ?

Section E style · Case study (4 marks)

Q10·4 marks·Case studyContinuity

Taxi fares. A taxi charges a flat ₹ $50$ for the first $2$ km and ₹ $15$ for each km after that, so the fare for $x$ km is $f (x) = 50$ for $0 < x \leq 2$ and $f (x) = 50 + 15 (x - 2)$ for $x > 2$ .

Is $f$ continuous at $x = 2$ ? [1 mark]
Is $f$ differentiable at $x = 2$ ? [1 mark]
Find the fare for a $10$ km ride and $f^{'} (6)$ . [2 marks]

Show worked answer

$f (2) = 50$ and $lim_{x \to 2^{+}} [50 + 15 (x - 2)] = 50$ : yes A1
Left derivative $0$ , right derivative $15$ : no A1
$f (10) = 50 + 15 \times 8 =$ ₹ $170$ M1
$f^{'} (6) = 15$ (₹ per km) A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can check continuity at a point, find k for continuity and differentiate with the chain rule, logs and exponentials.

Read first: 1. Continuity; 3. Rules 12 practice questions · checkpoint: 4 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

You can test differentiability, and use implicit, logarithmic, parametric and inverse-trig differentiation on board-standard questions.

Read first: 2. Differentiability; 3. Rules; 4. Inverse-trig simplification; Worked examples 1-3 16 practice questions · checkpoint: 4 questions, 12 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write long answers and case studies in full, especially 'prove that' second-derivative questions and xˣ-type derivatives.

Read first: 3. Rules; Worked examples 2-3; Board tips 6 practice questions · checkpoint: 3 questions, 13 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle piecewise functions with two unknowns, |x| and greatest-integer functions and max/min-defined functions.

Read first: 1. Continuity; 2. Differentiability; Common errors 6 practice questions · checkpoint: 3 questions, 13 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 6 of the 73 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceContinuity

The function $f (x) = {\begin{cases} \frac{\sin 5 x + \sin x}{3 x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$ . The value of $k$ is

(a) $\frac{1}{3}$
(b) $\frac{5}{3}$
(c) $2$
(d) $6$

Q2·1 mark·Assertion–reasonContinuity

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The function $f (x) = | \sin x |$ is continuous for all real $x$ .

Reason (R): If $g$ and $h$ are continuous functions, then the composite $g \circ h$ is continuous.

(a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c)Assertion (A) is true but Reason (R) is false.
(d)Assertion (A) is false but Reason (R) is true.

Q3·2 marks·Very short answerContinuity

Find the value of $k$ for which $f (x) = {\begin{cases} \frac{\sqrt{x + 7} - 3}{x - 2}, & x \neq 2 \\ k, & x = 2 \end{cases}$ is continuous at $x = 2$ .

Worked solution·2 marks·Very short answerParametric differentiation

For the curve $x = t - \frac{1}{t}$ , $y = t + \frac{1}{t}$ ( $t > 0$ ), find $\frac{d y}{d x}$ in terms of $t$ and its value at $t = 2$ .

Full worked solution Free, no sign-in

$\frac{d x}{d t} = 1 + \frac{1}{t^{2}}$ , $\frac{d y}{d t} = 1 - \frac{1}{t^{2}}$ , so $\frac{d y}{d x} = \frac{t^{2} - 1}{t^{2} + 1}$ A1
At $t = 2$ : $\frac{3}{5}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S130. All 24 →

Knowledge organiser: Continuity and Differentiability on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Derivative of sin x · Power rule. All formulas A–Z

Where marks are lost in Continuity and Differentiability

Continuity questions without LHL, RHL and f(c) written separately. Fix: write all three with their values; each carries a mark.
Calling a function differentiable because the pieces join up. Fix: continuity is necessary, not sufficient; compute LHD and RHD and compare.
Using the power rule or the exponential rule on xˣ or (sin x)ˣ. Fix: take logs first: log y = x log(sin x), then differentiate both sides.
Forgetting dy/dx on y-terms in implicit differentiation, or the inner derivative in the chain rule. Fix: underline each y-term and each inner function before differentiating.
Parametric second derivative as (d²y/dt²)/(d²x/dt²). Fix: d²y/dx² = d/dt(dy/dx) ÷ dx/dt.
Inverse-trig substitution without the interval. Fix: write 'x = cos θ, θ ∈ (0, π)' so the simplification is justified.
In 'prove that' second-order questions, differentiating a messy quotient. Fix: first clear fractions or roots (e.g. multiply out y₁√(1 − x²) = …), then differentiate.

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Examiner Insights: common mistakes in CBSE Class 12 Maths, with fixes (our analysis of public sources) →

Continuity and Differentiability in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 8, 9, 10, 22, 26
Sample paper 2 (Board standard): questions 8, 9, 10, 22, 26
Sample paper 3 (Above board): questions 8, 9, 22, 26, 27
Sample paper 4 (Above board): questions 8, 9, 10, 22, 26
Sample paper 5 (95+ challenge): questions 8, 9, 22, 26, 27

Want it against the clock? Take a timed 30-mark chapter test on Continuity and Differentiability, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on limits, continuity and differentiability.

Continuity and Differentiability: common questions

What is in CBSE Class 12 Maths Chapter 5, Continuity and Differentiability?

The topics are: Continuity, Differentiability, Chain rule and composite functions, Derivatives of inverse trigonometric functions, Exponential and logarithmic functions, Parametric differentiation, Second order derivatives, Logarithmic differentiation, Implicit differentiation. This page has revision notes for each, and 73 board-style practice questions with step-marked solutions.

How many marks is Continuity and Differentiability worth?

It is in the Calculus unit, which carries 35 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Continuity and Differentiability?

Continuity questions without LHL, RHL and f(c) written separately. Fix: write all three with their values; each carries a mark. Calling a function differentiable because the pieces join up. Fix: continuity is necessary, not sufficient; compute LHD and RHD and compare.

Are the Continuity and Differentiability important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Continuity and Differentiability — revision notes

1. Continuity

2. Differentiability

3. Rules

4. Inverse-trig simplification

Worked example 1

Worked example 2

Worked example 3

Common errors

Board tips

Continuity and Differentiability Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Continuity and Differentiability

Continuity and Differentiability in our sample papers

Continuity and Differentiability: common questions

Work through the Route to 95 with your progress saved