Class 12 · Chapter 6 · Calculus unit (35 of 80 marks)

Application of Derivatives Class 12: notes and important questions

Q: What is in CBSE Class 12 Maths Chapter 6, Application of Derivatives?

The topics are: Rate of change of quantities, Increasing and decreasing functions, Maxima and minima (local), Absolute maxima and minima on a closed interval, Optimisation problems. This page has revision notes for each, and 43 board-style practice questions with step-marked solutions.

Q: How many marks is Application of Derivatives worth?

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

Q: Where do students lose marks in Application of Derivatives?

Substituting the particular value before differentiating in a rate problem. Fix: differentiate the general relation with respect to t first, then substitute. Not justifying the maximum or minimum. Fix: show the second-derivative sign (or the first-derivative sign change); this carries a mark.

Q: Are the Application of Derivatives 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).

Q: 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.

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 43 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

43 questions
11 multiple choice, 3 assertion–reason, 9 very short answer, 10 short answer, 6 long answer, 4 case study
About 16 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 6 on ncert.nic.in (PDF)

Revision notes

Application of Derivatives — revision notes

1. Rate of change

$\frac{d y}{d x}$ is the rate of change of $y$ with respect to $x$ . If both depend on time, $\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}$ . Write a relation between the variables first, differentiate with respect to $t$ , and only then substitute the instantaneous values. In economics, marginal cost $= C^{'} (x)$ and marginal revenue $= R^{'} (x)$ .

2. Increasing and decreasing functions

$f^{'} > 0$ on an open interval $\Rightarrow f$ strictly increasing there; $f^{'} < 0 \Rightarrow$ strictly decreasing.
$f^{'} \geq 0$ with $f^{'} = 0$ only at isolated points still gives strictly increasing (e.g. $x^{3}$ ).
Method: find where $f^{'} = 0$ or $f^{'}$ is undefined, split the line, test the sign of $f^{'}$ in each part. Square factors such as $(x - 1)^{2}$ do not change sign.
Inequalities: to show $f (x) > g (x)$ for $x > a$ , prove $h = f - g$ is increasing and $h (a) \geq 0$ .

3. Local maxima and minima

Critical point: $f^{'} (c) = 0$ or $f^{'} (c)$ does not exist.
First derivative test: $f^{'}$ changes $+ \to -$ : local max; $- \to +$ : local min; no change: neither.
Second derivative test: $f^{'} (c) = 0$ and $f^{″} (c) < 0$ : local max; $f^{″} (c) > 0$ : local min; $f^{″} (c) = 0$ : test fails, use the first derivative test.

4. Absolute extrema on $[a, b]$

Evaluate $f$ at all critical points in $(a, b)$ and at the end points $a, b$ ; the largest value is the absolute maximum, the smallest the absolute minimum.

5. Optimisation strategy

Draw/label; write the quantity to optimise.
Use the constraint to reduce it to one variable; state the domain.
Differentiate, solve, and justify max/min (second derivative or sign change).
Answer the question actually asked (dimensions, cost, …) with units.

Worked example 1

A spherical balloon is inflated at $100 {cm}^{3} / s$ . When $r = 5$ cm: $\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t} \Rightarrow \frac{d r}{d t} = \frac{100}{100 π} = \frac{1}{π} cm/s$ .

Worked example 2

$f (x) = 2 x^{3} - 3 x^{2} - 12 x$ : $f^{'} (x) = 6 (x - 2) (x + 1)$ . Increasing on $(- \infty, - 1]$ and $[2, \infty)$ , decreasing on $[- 1, 2]$ ; local max $f (- 1) = 7$ , local min $f (2) = - 20$ .

Worked example 3

Of all rectangles with perimeter $40$ cm, the area $A = x (20 - x)$ is greatest when $A^{'} (x) = 20 - 2 x = 0$ , i.e. the square of side $10$ cm ( $A^{″} = - 2 < 0$ ).

Common errors

Substituting the particular value before differentiating in a rate problem.
Forgetting end points when finding absolute extrema; confusing a local maximum with the absolute maximum.
Ignoring points where $f^{'}$ is undefined (e.g. $x^{2 / 3}$ at $0$ ).
Not justifying that the stationary point is a maximum/minimum — this carries a mark.
Rejecting a variable value that is outside the domain without comment (e.g. negative lengths): say why it is rejected.

Board tips

Tangents and normals and approximations have been removed from the syllabus; questions come from rate of change, monotonicity and maxima/minima.
Write intervals clearly; either open or closed end points are accepted for strict monotonicity where $f$ is continuous.
In case-based questions, read what each part asks — part (iii) often needs the value (profit, area), not just $x$ .

Topics in this chapter: Rate of change of quantities · Increasing and decreasing functions · Maxima and minima (local) · Absolute maxima and minima on a closed interval · Optimisation problems.

Formulas for this chapter: Chapter 6 on the Class 12 formula sheet (free PDF and A3 poster).

Application of Derivatives 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 · 194 KB

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

Q1·1 mark·Multiple choiceIncreasing and decreasing functions

$f (x) = x^{3} - 3 x$ is decreasing on the interval

Show worked answer

$f^{'} (x) = 3 x^{2} - 3 = 3 (x - 1) (x + 1) < 0$ exactly when $- 1 < x < 1$ A1
Correct option: (a) $(- 1, 1)$

Q2·1 mark·Multiple choiceRate of change

How fast does the area of a circle change with its radius $r$ at the instant when $r = 7$ cm?

Q3·1 mark·Assertion–reasonMaxima and minima

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

Assertion (A): $x = 0$ is a point of local minimum of $f (x) = x^{3}$ .

Reason (R): For $f (x) = x^{3}$ , $f^{'} (0) = 0$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerRate of change

The edge of a cube is increasing at $3$ cm/s. How fast is its volume increasing when the edge is $10$ cm long?

Show worked answer

Answer:

900 {cm}^{3} / s

$\begin{aligned} V = x^{3} \\ \Rightarrow & \frac{d V}{d t} = 3 x^{2} \frac{d x}{d t} \end{aligned}$ M1
$= 3 \times 100 \times 3 = 900 {cm}^{3} / s$ A1

Q5·2 marks·Short answerIncreasing and decreasing functions

Find the intervals in which $f (x) = 2 x^{2} - 8 x + 5$ is increasing and those in which it is decreasing.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerMaxima and minima

Find the local maximum and local minimum values of $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ .

Show worked answer

Answer: Local max

6

x = 1

; local min

2

x = 3

$\begin{aligned} f^{'} (x) = 3 x^{2} - 12 x + 9 = 3 (x - 1) (x - 3) = 0 \\ \Rightarrow & x = 1, 3 \end{aligned}$ M1
$f^{″} (x) = 6 x - 12$ : $f^{″} (1) = - 6 < 0$ , local maximum $f (1) = 6$ A1
$f^{″} (3) = 6 > 0$ , local minimum $f (3) = 2$ A1

Q7·3 marks·Short answerIncreasing and decreasing functions

On which intervals is $f (x) = x^{4} - 8 x^{2}$ strictly increasing, and on which is it strictly decreasing?

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerOptimisation problems

A rectangular garden of area $200 m^{2}$ is to be fenced on three sides; the fourth side is a wall. Find the dimensions that need the least fencing, and that length of fencing.

Show worked answer

Answer:

10

m by

20

m (the

20

m side parallel to the wall);

40

Let the two sides at right angles to the wall be $x$ m and the side parallel to it $y$ m: $x y = 200$ M1
Fencing $L = 2 x + y = 2 x + \frac{200}{x}$ A1
$\begin{aligned} \frac{d L}{d x} = 2 - \frac{200}{x^{2}} = 0 \\ \Rightarrow & x^{2} = 100 \\ \Rightarrow & x = 10 \end{aligned}$ M1
$\frac{d^{2} L}{d x^{2}} = \frac{400}{x^{3}} > 0$ , so this is a minimum A1
$x = 10$ m, $y = 20$ m; least fencing $= 40$ m A1

Q9·5 marks·Long answerOptimisation problems

A closed cylindrical can must hold $128 π {cm}^{3}$ . Find the radius and height that use the least metal (least total surface area), and that surface area.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyOptimisation problems

Workshop profit. A workshop's revenue from making $x$ chairs a week is $R (x) = 120 x - 2 x^{2}$ and its cost is $C (x) = 20 x + 400$ (both in ₹ hundreds), so its profit is $P (x) = R (x) - C (x)$ .

Find $P^{'} (x)$ . [1 mark]
How many chairs give the greatest profit? [1 mark]
Find the greatest profit and show that it is a maximum. [2 marks]

Show worked answer

$P (x) = 100 x - 2 x^{2} - 400$ , so $P^{'} (x) = 100 - 4 x$ A1
$100 - 4 x = 0 \Rightarrow x = 25$ A1
$P^{″} (x) = - 4 < 0$ , so $x = 25$ gives a maximum M1
$P (25) = 2500 - 1250 - 400 = 850$ , i.e. ₹ $85000$ 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 find a rate of change, find where a function increases or decreases and locate turning points.

Read first: 1. Rate of change; 2. Increasing and decreasing functions 12 practice questions · checkpoint: 4 questions, 4 marks, pass 80%

Practise step 1

Step 2

Board standard

You can classify local maxima and minima with a derivative test and find absolute extrema on a closed interval.

Read first: 2. Increasing and decreasing functions; 3. Local maxima and minima; 4. Absolute extrema on [a, b]; Worked examples 1-2 16 practice questions · checkpoint: 4 questions, 9 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write full-marks optimisation and case-study answers: function set up, derivative, test, and the value that was asked for.

Read first: 5. Optimisation strategy; Worked example 3; Board tips 6 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle monotonicity with parameters or trig functions and the harder optimisation problems.

Read first: 2. Increasing and decreasing functions; 5. Optimisation strategy; 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. 16 of the 43 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceRate of change of quantities

The area of a circular oil patch on water is increasing at $48 π {cm}^{2} / s$ . At the instant when its radius is $8$ cm, the radius is increasing at

(a) $6 cm/s$
(b) $3 cm/s$
(c) $\frac{3}{2} cm/s$
(d) $12 cm/s$

Q2·1 mark·Multiple choiceRate of change of quantities

A shop’s revenue (in ₹) from selling $x$ units of a product is $R (x) = 30 x - 0.01 x^{2}$ . The marginal revenue when $x = 500$ is

(a)₹ $20$
(b)₹ $25$
(c)₹ $10$
(d)₹ $12 500$

Q3·1 mark·Multiple choiceIncreasing and decreasing functions

The function $f (x) = x^{3} - 12 x$ is strictly decreasing on

(a) $(- \infty, 2)$
(b) $(- \infty, - 2)$
(c) $(2, \infty)$
(d) $(- 2, 2)$

Worked solution·2 marks·Very short answerRate of change of quantities

The height of a closed cylindrical container is always twice its radius. If the radius is increasing at $0.1$ cm/s, find the rate at which its volume is increasing when the radius is $3$ cm.

Full worked solution Free, no sign-in

$V = π r^{2} (2 r) = 2 π r^{3} \Rightarrow \frac{d V}{d t} = 6 π r^{2} \frac{d r}{d t}$ M1
$= 6 π (9) (0.1) = 5.4 π {cm}^{3} / s$ 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 S131. All 24 →

Knowledge organiser: Application of Derivatives on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Application of Derivatives

Substituting the particular value before differentiating in a rate problem. Fix: differentiate the general relation with respect to t first, then substitute.
Not justifying the maximum or minimum. Fix: show the second-derivative sign (or the first-derivative sign change); this carries a mark.
Forgetting the end points when finding absolute extrema. Fix: tabulate f at every critical point and both end points, then choose.
Ignoring points where f′ is undefined (e.g. x^(2/3) at 0). Fix: list 'f′ = 0 or f′ undefined' when finding critical points.
Case-study part (iii) answered with x instead of the maximum value (profit, area). Fix: underline what is asked and finish with that quantity and its units.
Rejecting a stationary value outside the domain without comment. Fix: write why it is rejected (e.g. length must be positive).
Studying tangents, normals or approximations. Fix: they are out of the current syllabus; spend the time on rates, monotonicity and extrema.

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

Application of Derivatives 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 11, 23, 37
Sample paper 2 (Board standard): questions 11, 12, 23, 33
Sample paper 3 (Above board): questions 10, 23, 37
Sample paper 4 (Above board): questions 11, 12, 23, 29, 33
Sample paper 5 (95+ challenge): questions 10, 33, 37

Want it against the clock? Take a timed 30-mark chapter test on Application of Derivatives, 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 application of derivatives.

Application of Derivatives: common questions

What is in CBSE Class 12 Maths Chapter 6, Application of Derivatives?

The topics are: Rate of change of quantities, Increasing and decreasing functions, Maxima and minima (local), Absolute maxima and minima on a closed interval, Optimisation problems. This page has revision notes for each, and 43 board-style practice questions with step-marked solutions.

How many marks is Application of Derivatives 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 Application of Derivatives?

Substituting the particular value before differentiating in a rate problem. Fix: differentiate the general relation with respect to t first, then substitute. Not justifying the maximum or minimum. Fix: show the second-derivative sign (or the first-derivative sign change); this carries a mark.

Are the Application of Derivatives 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.

Work through the Route to 95 with your progress saved

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Revision notes

Application of Derivatives — revision notes

1. Rate of change

2. Increasing and decreasing functions

3. Local maxima and minima

4. Absolute extrema on [a,b]

5. Optimisation strategy

Worked example 1

Worked example 2

Worked example 3

Common errors

Board tips

Application of Derivatives 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 Application of Derivatives

Application of Derivatives in our sample papers

Application of Derivatives: common questions

Work through the Route to 95 with your progress saved

4. Absolute extrema on $[a, b]$