Class 12 Maths NCERT Solutions Chapter 6 Application of Derivatives | Boundless Maths
Unit III · Calculus · Chapter 6

Class 12 Maths NCERT Solutions Chapter 6: Application of Derivatives

Free, step-by-step NCERT Solutions for all four parts of this chapter — rate of change of quantities, increasing and decreasing functions, maxima and minima, and absolute maximum/minimum on a closed interval. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

4Exercises (incl. Misc.)
82Total Questions
2026-27CBSE Syllabus
100%Solved

Class 12 Maths NCERT Solutions Chapter 6 — Overview

This Class 12 Maths NCERT Solutions Chapter 6 hub covers Application of Derivatives, which takes the differentiation techniques from Chapter 5 and turns them into a practical toolkit. You'll use the derivative to measure how fast one quantity changes as another one does, to work out where a function is rising or falling, and — most importantly for the board exam — to find the largest or smallest value a function can take, whether that's the maximum area you can enclose, the minimum material needed for a box, or the shortest possible distance between two points.

The chapter builds in a deliberate sequence: rate of change first (pure differentiation, applied), then increasing/decreasing behaviour (reading the sign of the derivative), then maxima and minima (using that sign behaviour to locate turning points), and finally absolute maxima and minima on a closed interval, which is what most real optimisation problems — the ones that show up as 5-mark or case-study questions — actually need.

How the Chapter Builds

One Idea Leads to the Next

1

Rate of Change

dy/dx tells you how fast y changes as x changes. Exercise 6.1.

→
2

Increasing / Decreasing

The sign of f′(x) tells you whether the function is rising or falling. Exercise 6.2.

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3

Maxima & Minima

Where f′(x) changes sign (or f″(c) tells you the curve's shape), you get a turning point. Exercise 6.3.

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4

Absolute Max/Min

On a closed interval, compare critical points with the endpoints to find the true largest/smallest value. Misc. Exercise.

Quick Reference

Important Formulas — Chapter 6

Everything you need before you start solving. This is a summary for quick recall — the Formula Deck below has the full printable version for all of Calculus.

Rate of Change (§6.2)

Rate of change of y w.r.t. x

\dfrac{dy}{dx} = f'(x)

Rate of change of y at a specific point: \left.\dfrac{dy}{dx}\right|_{x=x_0}

Chain Rule (two quantities varying with a third)

\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt},\quad \dfrac{dx}{dt}\neq 0

Essential whenever both x and y depend on time t.

Mensuration formulas you'll need constantly

A=\pi r^2,\; C=2\pi r,\; V_{cube}=x^3,\; S_{cube}=6x^2,\; V_{sphere}=\tfrac{4}{3}\pi r^3

Nearly every question in Ex 6.1 hides a geometry formula — keep these on hand.

Increasing / Decreasing Functions (§6.3)

Increasing on an interval I

f'(x) \geq 0 \;\; \forall\, x \in I

Strictly increasing if f′(x) > 0 throughout I.

Decreasing on an interval I

f'(x) \leq 0 \;\; \forall\, x \in I

Strictly decreasing if f′(x) < 0 throughout I.

Maxima and Minima (§6.4)

Critical point

f'(c) = 0 \;\; \text{or}\;\; f \text{ not differentiable at } c

Necessary condition for a local max/min — not sufficient by itself.

First Derivative Test

Sign of f′(x): + \to - = local max; - \to + = local min; no change = point of inflexion

Works even where f is not twice differentiable.

Second Derivative Test

f'(c)=0,\; f''(c)<0 \Rightarrow \text{local max}
f'(c)=0,\; f''(c)>0 \Rightarrow \text{local min}

If f″(c) = 0 too, the test fails — go back to the first derivative test.

Absolute Maximum/Minimum on [a, b]

Working rule

1) Find critical points in (a,b)   2) Include endpoints a, b   3) Evaluate f at all of them   4) Largest = absolute max, smallest = absolute min

Missing the endpoints is the single most common way marks are lost here.

Decision Guide

Which Test Should I Use?

A quick way to decide, once you've found where f′(x) = 0.

SituationUse thisWhy
f is easy to differentiate twice, and f″(c) ≠ 0Second Derivative TestFastest — one substitution tells you max or min.
f″(c) = 0, or f isn't differentiable at cFirst Derivative TestSecond derivative test explicitly fails here — sign-check f′(x) on either side instead.
The problem gives a closed interval [a, b]Absolute Max/Min working ruleLocal tests alone can miss the true maximum, which may sit at an endpoint.
You just need to prove f is increasing/decreasing (no turning point needed)Sign of f′(x) directlyNo need to locate critical points at all for a pure increasing/decreasing proof.
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all four exercises.

  • Forgetting the Chain Rule when two quantities both vary with time — differentiating V = x³ directly instead of going through dV/dt = 3x²·(dx/dt).
  • Treating every critical point as an extremum. f′(c) = 0 is necessary, not sufficient — f(x) = x³ has f′(0) = 0 but x = 0 is a point of inflexion, not a max or min.
  • Skipping the endpoints when finding absolute max/min on a closed interval — the true maximum or minimum often sits at a or b, not at a critical point.
  • Using the second derivative test when it has already failed (f″(c) = 0) instead of switching to the first derivative test.
  • Sign errors while testing intervals — picking a test point outside the interval, or misjudging the sign of a product like (x−1)(x+2).
  • Ignoring domain restrictions in optimisation problems — forgetting that a length, radius or edge can't be negative, and keeping an invalid root.
  • Confusing the point of maxima with the maximum value — stating x = c as the answer when the question asks for f(c), or vice versa.
  • Not stating continuity explicitly at interval boundaries when proving increasing/decreasing behaviour — CBSE's marking scheme awards a step for this.
Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 6 — Choose an Exercise

6.1

Exercise 6.1

Rate of change of quantities — circles, cubes, spheres, cones, marginal cost & revenue · 18 questions

Solve Exercise 6.1 →
6.2

Exercise 6.2

Increasing and decreasing functions — proving monotonicity and finding intervals · 19 questions

Solve Exercise 6.2 →
6.3

Exercise 6.3

Maxima and minima — local extrema, first & second derivative test, absolute max/min, optimisation · 29 questions

Solve Exercise 6.3 →
M

Miscellaneous Exercise

Mixed application questions combining rate of change, monotonicity and optimisation · 16 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Calculus — differentiation, applications of derivatives, integration — in one printable PDF.

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Common Questions

Frequently Asked Questions

Quick answers about Chapter 6, Application of Derivatives.

How many exercises are there in Chapter 6, Application of Derivatives?
There are three main exercises — 6.1, 6.2 and 6.3 — plus a Miscellaneous Exercise, totalling 82 questions across rate of change, increasing/decreasing functions, and maxima-minima.
Is this chapter important for the board exam?
Yes — it's one of the highest-weightage chapters in the Calculus unit, and the maxima-minima (optimisation) section is a near-certain contributor to Section D or Section E most years.
What is the difference between the first derivative test and the second derivative test?
The first derivative test checks whether f′(x) changes sign around a critical point. The second derivative test checks the sign of f″(c) instead, which is usually faster — but it fails when f″(c) = 0, and you must fall back to the first derivative test.
What should I revise before starting this chapter?
Make sure Chapter 5 (Continuity and Differentiability) is solid first — every method in this chapter depends on differentiating functions quickly and correctly, including implicit and chain-rule differentiation.
Where can I find the official NCERT textbook for this chapter?
Application of Derivatives is Chapter 6 of the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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