Class 12 Maths Chapter 6: Application of Derivatives

Derivatives are not just mathematical concepts—they are powerful tools used to analyze change, optimize outcomes, and solve practical problems in science, engineering, economics, and everyday life. The Application of Derivatives chapter demonstrates how differentiation can be used to study increasing and decreasing functions, determine maxima and minima, calculate rates of change, and solve optimization problems. It is one of the highest-weightage chapters for CBSE Board, JEE Main, and JEE Advanced, with a strong focus on conceptual understanding and problem-solving.

At ALLEN, Chapter 6: Application of Derivatives smart study materials are designed to help you master the Application of Derivatives through concise theory, essential formulas, shortcut techniques, solved illustrations, and exam-oriented insights. Whether you're preparing for board exams or competitive entrance tests, these notes provide everything required for quick and effective revision.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	3–4 Hours
Key Concepts Covered	First Derivative Test, Second Derivative Test, Maxima & Minima, Approximation using Differentials

2.0Chapter 6 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Test your Knowledge

question 1 of 5

The derivative of a function at a point represents the:

1.Area under the curve

2.Slope of the tangent

3.Maximum value of the function

4.Length of the curve

Must-Revise Concepts

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Home

JEE Maths

Class 12 Chapter 6

Class 12 Maths Chapter 6: Application of Derivatives

Derivatives are not just mathematical concepts—they are powerful tools used to analyze change, optimize outcomes, and solve practical problems in science, engineering, economics, and everyday life. The Application of Derivatives chapter demonstrates how differentiation can be used to study increasing and decreasing functions, determine maxima and minima, calculate rates of change, and solve optimization problems. It is one of the highest-weightage chapters for CBSE Board, JEE Main, and JEE Advanced, with a strong focus on conceptual understanding and problem-solving.

At ALLEN, Chapter 6: Application of Derivatives smart study materials are designed to help you master the Application of Derivatives through concise theory, essential formulas, shortcut techniques, solved illustrations, and exam-oriented insights. Whether you're preparing for board exams or competitive entrance tests, these notes provide everything required for quick and effective revision.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate to High
Revision Time	3–4 Hours
Key Concepts Covered	First Derivative Test, Second Derivative Test, Maxima & Minima, Approximation using Differentials

2.0Chapter 6 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Test your Knowledge

question 1 of 5

The derivative of a function at a point represents the:

1.Area under the curve

2.Slope of the tangent

3.Maximum value of the function

4.Length of the curve

Must-Revise Concepts

Applications of Derivatives

The applications of derivatives is a core topic in calculus and mathematical analysis. It primarily involves using derivatives to understand and solve various mathematical problems. Key mathematical applications include:

Finding Maxima and Minima: Derivatives help identify the local maxima and minima of a function. This is done by finding the critical points where the first derivative f'(x) is zero or undefined and then using the second derivative f''(x) to determine whether the point is a maximum or minimum.
Rate of Change of Quantities: The derivative dy/dx measures the instantaneous rate of change of a dependent variable y with respect to an independent variable x. This concept is used in problems involving velocity, acceleration, and other changing quantities.
Increasing and Decreasing Functions: A function f(x) is increasing on an interval if f'(x) > 0 for all x in that interval and decreasing if f'(x) < 0. This helps in analyzing the monotonicity of functions.
Finding Tangents and Normal: The derivative f'(x) at a point gives the slope of the tangent to the curve at that point. The equation of the tangent line can be written as y – y₁ = f'(x₁)(x – x₁), and the normal line is perpendicular to the tangent.

1.0Rate of Change of Quantities

The rate of change of quantities is a fundamental concept in calculus and is a direct application of derivatives. It is used to measure how one quantity changes with respect to another. Understanding this concept allows students and professionals to solve a variety of real-world problems, such as determining how fast a car is accelerating, how the height of a balloon increases with time, or how a population changes over a certain period.

In mathematical terms, if y = f(x) is a function representing the relationship between y and x, then the derivative $\frac{d y}{d x}$ signifies the rate of change of y with respect to x. Here, x could represent time, distance, or any other independent variable, while y could represent a dependent variable like speed, volume, or temperature.

Mathematical Definition

The rate of change of a quantity y with respect to x is defined as: $\frac{d y}{d x} = lim_{Δ x \to 0} \frac{Δ y}{Δ x}$

where:

$Δ y$ is the change in y.
$Δ x$ is the change in x.

The derivative $\frac{d y}{d x}$ tells us how much y changes when x changes by a small amount. If $\frac{d y}{d x} > 0$ , then y is increasing with respect to x. If $\frac{d y}{d x} < 0$ , then y is decreasing.

2.0Increasing and Decreasing Functions

In mathematics, particularly in calculus, functions can be classified based on their behavior over certain intervals.

Increasing Functions: A function f(x) is said to be increasing on an interval if, for any two points x₁ and x₂ within that interval where x₁ < x₂, the function values satisfy $f (x_{1}) \leq f (x_{2})$ . This means that as x increases, f(x) either increases or remains constant.
Decreasing Functions: Conversely, a function f(x) is decreasing on an interval if, for any two points x₁and x₂within that interval where x₁ < x₂, the function values satisfy $f (x_{1}) \geq f (x_{2})$ . This means that as ( x ) increases, ( f(x) ) either decreases or remains constant.

Understanding whether a function is increasing or decreasing is crucial for analyzing its behavior, finding local maxima and minima, and solving optimization problems.

1.0Maxima and Minima

Definitions:

Local Maximum: A function f(x) has a local maximum at x = a if f(a) is greater than or equal to f(x) for all x in some open interval around a. This means f(a) is the highest point in a small neighbourhood around a.
Local Minimum: A function f(x) has a local minimum at x = b if f(b) is less than or equal to f(x) for all x in some open interval around b. This means f(b) is the lowest point in a small neighbourhood around b.
Absolute Maximum: A function f(x) has an absolute maximum at x = c if f(c) is greater than or equal to f(x) for all x in the domain of f. This means f(c) is the highest point over the entire domain.
Absolute Minimum: A function f(x) has an absolute minimum at x = d if f(d) is less than or equal to f(x) for all x in the domain of f. This means f(d) is the lowest point over the entire domain.

Finding Maxima and Minima:

To find the maxima and minima of a function, we typically follow these steps:

Step 1: Find the First Derivative: Compute the first derivative of the function.

Step 2: Find Critical Points: Set the first derivative equal to zero and solve for x. These values are called critical points. Also, include points where the derivative does not exist.

Step 3: Determine the Nature of Critical Points:

Second Derivative Test: Compute the second derivative, f’'(x). Evaluate the second derivative at each critical point.

If f’'(x) > 0 at a critical point, the function has a local minimum at that point.

If f’'(x) < 0 at a critical point, the function has a local maximum at that point.

If f’'(x) = 0, the test is inconclusive.

First Derivative Test: Alternatively, examine the sign of the first derivative before and after each critical point.

If f’(x) changes from positive to negative at a critical point, the function has a local maximum there.

If f’(x) changes from negative to positive at a critical point, the function has a local minimum there.

2.0Solved Examples on Rate of Change of Quantities

Example 1: A stone is dropped into a calm lake, creating circular waves that expand at a speed of 4 cm per second. At the moment when the radius of the circular wave reaches 10 cm, how quickly is the enclosed area increasing?

Solution:

The area A of a circle with radius r is given by $A = π r^{2}$ .To find the rate of change of the area A with respect to time t, we differentiate:

$\frac{d A}{d t} = \frac{d}{d t} (π r^{2})$

$\frac{d A}{d t} = π \frac{d}{d t} (r^{2})$

$\frac{d A}{d t} = π (2 r \frac{d r}{d t})$

$\frac{d A}{d t} = 2 π r \frac{d r}{d t}$

Given that:

$\frac{d r}{d t} = 4 cm / s$

Therefore:

$\frac{d A}{d t} = 2 π (10) (4) = 80 π$

Thus, the enclosed area is increasing at a rate of $80 π cm^{2} / s$ when r = 10 cm.

Example 2: A balloon is being inflated and its volume is increasing at a constant rate of 5 cubic centimeters per second. At what rate is the radius of the balloon expanding when the radius is 6 cm?

Solution:

The volume V of a sphere with radius r is given by $V = \frac{4}{3} π r^{3}$ . To find the rate of change of the radius r with respect to time t, we differentiate the volume formula with respect to time:

$\frac{d V}{d t} = \frac{d}{d t} (\frac{4}{3} π r^{3})$

$\frac{d V}{d t} = \frac{4}{3} π \cdot 3 r^{2} \frac{d r}{d t}$

$\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$

Given that:

$\frac{d V}{d t} = 5 cm^{3} / s$

We need to find $\frac{d r}{d t}$ when r = 6 cm. Substituting the known values into the differentiated equation:

$5 = 4 π (6)^{2} \frac{d r}{d t}$

$5 = 144 π \frac{d r}{d t}$

$\frac{d r}{d t} = \frac{5}{144 π}$

Therefore, the radius is increasing at a rate of $\frac{5}{144 π} cm / s$ when the radius is 6 cm.

Example 3: Consider the function $f (x) = x^{3} - 3 x^{2} + 4.$

Solution:

Let $f (x) = x^{3} - 3 x^{2} + 4$

Step 1: Find the first derivative: f’(x) = 3x² – 6x

Step 2: Find critical points by setting the first derivative to zero:

⇒ 3x² – 6x = 0

⇒ 3x (x – 2) = 0

⇒ x = 0 or x = 2

Step 3: Use the second derivative test: f’'(x) = 6x – 6

⇒ At x = 0: f’'(0) = 6(0) – 6 = –6 (negative, so local maximum)

⇒ At x = 2: f’'(2) = 6(2) – 6 = 6 (positive, so local minimum)

Therefore, x = 0 is a local maximum and x = 2 is a local minimum.

Example 4: Consider the function $g (x) = x^{4} - 4 x^{3} + 6 x^{2} .$

Solution:

Let $g (x) = x^{4} - 4 x^{3} + 6 x^{2}$

Step 1: Find the first derivative: $g^{'} (x) = 4 x^{3} - 12 x^{2} + 12 x$

Step 2: Find critical points by setting the first derivative to zero:

$\Rightarrow 4 x (x^{2} - 3 x + 3) = 0$

$\Rightarrow x = 0 or solve x^{2} - 3 x + 3 = 0$

Step 3: Use the second derivative test: $g^{''} (x) = 12 x^{2} - 24 x + 12$

⇒ At x = 0: $g^{''} (0) = 12 (0)^{2} - 24 (0) + 12 = 12$ (positive, so local minimum)

Therefore, x = 0 is a local minimum. The quadratic has no real roots, so there are no other critical points.

Example 5: Show that the function $f (x) = 3 x^{2} - 6 x + 4$ is increasing on the set of real numbers $R$ .

Solution:

To determine if the function $f (x) = 3 x^{2} - 6 x + 4$ is increasing on $R$ , we need to find its first derivative and analyze its sign.

Find the first derivative: $f^{'} (x) = \frac{d}{d x} (3 x^{2} - 6 x + 4) = 6 x - 6$
Simplify the derivative: $f ’ (x) = 6 (x -1)$
Analyze the sign of the derivative: Since 6(x – 1) is always positive for all $x \in R$ , the function f(x) is increasing on $R$ .

Therefore, the function $f (x) = 3 x^{2} - 6 x + 4$ is increasing on the set of real numbers $R$ .

Example 6: Determine the intervals on which the function $f (x) = sin x - cos x$ is increasing or decreasing for $0 \leq x \leq 2 π$

Solution:

To find the intervals where the function $f (x) = sin x - cos x$ is increasing or decreasing, we need to analyze its first derivative.

Find the first derivative: $f^{'} (x) = cos x + sin x$
Determine where f’(x) is positive or negative: $f^{'} (x) > 0 \Rightarrow cos x + sin x > 0$

$f^{'} (x) < 0 \Rightarrow cos x + sin x < 0$

Solve the inequality: The points $\frac{π}{4}$ and $\frac{5 π}{4}$ divide the interval [0,2 \pi] into three subintervals:

$[0, \frac{π}{4}], [\frac{π}{4}, \frac{5 π}{4}]$ and $[\frac{5 π}{4}, 2 π]$

For $0 < x < \frac{π}{4}, cos x + sin x > 0$ , so f(x) is increasing.
For $\frac{π}{4} < x < \frac{5 π}{4}, cos x + sin x < 0$ , so f(x) is decreasing.
For $\frac{5 π}{4} < x < 2 π, cos x + sin x > 0$ , so f(x) is increasing.

Therefore, the function $f (x) = sin x - cos x$ is increasing on $[0, \frac{π}{4}]$ and $[\frac{5 π}{4}, 2 π]$ and decreasing on $[\frac{π}{4}, \frac{5 π}{4}]$ .

3.0Practice Questions on Applications of Derivatives

A spherical balloon is being inflated, causing its volume to increase at a rate of 10 cubic centimeters per second. Determine how quickly the radius of the balloon is growing when the radius measures 5 cm.
The radius of a circular oil spill is increasing at a rate of 0.1 meters per minute. How fast is the area of the oil spill increasing when the radius is 20 meters?
A conical tank is being filled with water at a speed of 5 cubic meters per minute. The tank has a height of 10 meters and a base radius of 3 meters. How fast is the water level rising when the water is 6 meters deep?

4.0Sample Questions on Applications of Derivatives

What is meant by the rate of change of a function?

Ans: The rate of change of a function y = f(x) w.r.t x is the derivative $\frac{d y}{d x}$ . It measures how y changes as x varies. For example, in physics, if s(t) represents displacement as a function of time t, then $\frac{d s}{d t}$ gives the velocity.

How do you find the equation of a tangent line using derivatives?

Ans: To find the tangent line to a curve y = f(x) at a point (x₁, y₁):

Find the derivative f'(x), which gives the slope of the tangent at any point x.
Evaluate f'(x) at x₁ to get the slope m.
Use the point-slope form to express the equation of a line: $y - y_{1} = m (x - x_{1})$

1.0Learning Outcomes

After completing this chapter, you will be able to:

Interpret the geometric meaning of derivatives.
Determine intervals where functions are increasing or decreasing.
Find local maxima and minima of functions.
Solve optimization problems involving maxima and minima.
Apply derivatives to calculate rates of change.
Analyze tangents and normals to curves.
Solve application-based problems for JEE and CBSE exams.

2.05-Minute Quick Revision

Must-Revise Concepts

Increasing and Decreasing Functions
Monotonicity
First Derivative Test
Second Derivative Test
Local Maxima and Minima
Absolute Maxima and Minima
Tangents and Normals
Rate of Change
Approximation using Differentials
Optimization Problems

Important Formula Recall

(f'(x)>0) ⇒ Increasing Function
(f'(x)<0) ⇒ Decreasing Function
Critical Point: (f'(x)=0) or undefined
Maximum/Minimum Conditions
Slope of Tangent = (f'(x))
Slope of Normal = (-\frac{1}{f'(x)})
Linear Approximation: dy=f'(x),dx

Important Properties / Shortcut Tips

Always identify the critical points before applying derivative tests.
Use the first derivative test to determine increasing and decreasing intervals.
Use the second derivative test only after finding stationary points.
Carefully interpret the question before choosing the optimization variable.
Draw rough graphs whenever possible to visualize maxima, minima, and monotonicity.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Increasing & Decreasing Functions	High	Easy–Moderate
First Derivative Test	Very High	Moderate
Second Derivative Test	Very High	Moderate
Maxima & Minima	Very High	Moderate–High
Tangents & Normals	High	Moderate
Rate of Change	High	Easy–Moderate
Optimization Problems	Very High	High
Approximation using Differentials	Moderate	Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Application of Derivatives in CBSE Board, JEE Main, and JEE Advanced examinations.

5.0Common Mistakes & JEE Tips

Students often lose marks in Application of Derivatives due to incorrect interpretation of derivative tests and optimization conditions.

Common Mistake	Correct Approach
Ignoring critical points	Find all points where (f'(x)=0) or undefined before testing.
Using only the second derivative test	Apply the first derivative test when the second derivative is inconclusive.
Incorrect sign analysis	Check the sign of (f'(x)) carefully in each interval.
Wrong slope of the normal	Use (m=-\frac{1}{f'(x)}), not (f'(x)).
Choosing the wrong optimization variable	Define variables and constraints before differentiating.
Ignoring the domain of the function	Verify that the obtained solution lies within the valid domain.
Algebraic errors after differentiation	Simplify expressions carefully before solving.
Forgetting endpoint values	Compare endpoint values whenever absolute maxima or minima are required.

6.0ALLEN Faculty Tips

Master Continuity and Differentiability before starting this chapter.
Practice sign chart questions for increasing and decreasing functions.
Learn both the first and second derivative tests thoroughly.
Draw rough graphs to understand maxima, minima, and monotonicity.
Practice optimization questions from different real-life scenarios.
Revise tangent and normal formulas regularly.
Solve Previous Years' Questions (PYQs) to improve speed and accuracy.

7.0PYQ Trend Analysis

Application of Derivatives is one of the highest-weightage chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are frequently asked on monotonicity, maxima and minima, tangents, normals, rate of change, and optimization.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Increasing & Decreasing Functions	1–2	Moderate
Maxima & Minima	2–3	Moderate to Difficult
First & Second Derivative Tests	1–2	Moderate
Tangents & Normals	1	Moderate
Rate of Change	1	Easy to Moderate
Optimization Problems	2–3	Moderate to Difficult

8.0Most Common Question Types

Finding intervals where a function is increasing or decreasing.
Determining local and absolute maxima or minima.
Applying first and second derivative tests.
Finding equations of tangents and normals.
Solving optimization problems involving area, volume, and cost.
Calculating rates of change in practical situations.
Identifying stationary points and points of inflection.
Solving graph-based and application-oriented questions.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise derivative formulas from the previous chapter.
Practice increasing and decreasing function problems.
Learn the first and second derivative tests.
Solve maxima and minima questions.
Practice tangent, normal, and rate of change problems.
Revise optimization techniques with different examples.
Solve Previous Years' Questions (PYQs).
Finish with a timed chapter-wise mock test to improve accuracy.

Class 12 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Applications of Derivatives

The applications of derivatives is a core topic in calculus and mathematical analysis. It primarily involves using derivatives to understand and solve various mathematical problems. Key mathematical applications include:

Finding Maxima and Minima: Derivatives help identify the local maxima and minima of a function. This is done by finding the critical points where the first derivative f'(x) is zero or undefined and then using the second derivative f''(x) to determine whether the point is a maximum or minimum.
Rate of Change of Quantities: The derivative dy/dx measures the instantaneous rate of change of a dependent variable y with respect to an independent variable x. This concept is used in problems involving velocity, acceleration, and other changing quantities.
Increasing and Decreasing Functions: A function f(x) is increasing on an interval if f'(x) > 0 for all x in that interval and decreasing if f'(x) < 0. This helps in analyzing the monotonicity of functions.
Finding Tangents and Normal: The derivative f'(x) at a point gives the slope of the tangent to the curve at that point. The equation of the tangent line can be written as y – y₁ = f'(x₁)(x – x₁), and the normal line is perpendicular to the tangent.

1.0Rate of Change of Quantities

The rate of change of quantities is a fundamental concept in calculus and is a direct application of derivatives. It is used to measure how one quantity changes with respect to another. Understanding this concept allows students and professionals to solve a variety of real-world problems, such as determining how fast a car is accelerating, how the height of a balloon increases with time, or how a population changes over a certain period.

In mathematical terms, if y = f(x) is a function representing the relationship between y and x, then the derivative $\frac{d y}{d x}$ signifies the rate of change of y with respect to x. Here, x could represent time, distance, or any other independent variable, while y could represent a dependent variable like speed, volume, or temperature.

Mathematical Definition

The rate of change of a quantity y with respect to x is defined as: $\frac{d y}{d x} = lim_{Δ x \to 0} \frac{Δ y}{Δ x}$

where:

$Δ y$ is the change in y.
$Δ x$ is the change in x.

The derivative $\frac{d y}{d x}$ tells us how much y changes when x changes by a small amount. If $\frac{d y}{d x} > 0$ , then y is increasing with respect to x. If $\frac{d y}{d x} < 0$ , then y is decreasing.

2.0Increasing and Decreasing Functions

In mathematics, particularly in calculus, functions can be classified based on their behavior over certain intervals.

Increasing Functions: A function f(x) is said to be increasing on an interval if, for any two points x₁ and x₂ within that interval where x₁ < x₂, the function values satisfy $f (x_{1}) \leq f (x_{2})$ . This means that as x increases, f(x) either increases or remains constant.
Decreasing Functions: Conversely, a function f(x) is decreasing on an interval if, for any two points x₁and x₂within that interval where x₁ < x₂, the function values satisfy $f (x_{1}) \geq f (x_{2})$ . This means that as ( x ) increases, ( f(x) ) either decreases or remains constant.

Understanding whether a function is increasing or decreasing is crucial for analyzing its behavior, finding local maxima and minima, and solving optimization problems.

1.0Maxima and Minima

Definitions:

Local Maximum: A function f(x) has a local maximum at x = a if f(a) is greater than or equal to f(x) for all x in some open interval around a. This means f(a) is the highest point in a small neighbourhood around a.
Local Minimum: A function f(x) has a local minimum at x = b if f(b) is less than or equal to f(x) for all x in some open interval around b. This means f(b) is the lowest point in a small neighbourhood around b.
Absolute Maximum: A function f(x) has an absolute maximum at x = c if f(c) is greater than or equal to f(x) for all x in the domain of f. This means f(c) is the highest point over the entire domain.
Absolute Minimum: A function f(x) has an absolute minimum at x = d if f(d) is less than or equal to f(x) for all x in the domain of f. This means f(d) is the lowest point over the entire domain.

Finding Maxima and Minima:

To find the maxima and minima of a function, we typically follow these steps:

Step 1: Find the First Derivative: Compute the first derivative of the function.

Step 2: Find Critical Points: Set the first derivative equal to zero and solve for x. These values are called critical points. Also, include points where the derivative does not exist.

Step 3: Determine the Nature of Critical Points:

Second Derivative Test: Compute the second derivative, f’'(x). Evaluate the second derivative at each critical point.

If f’'(x) > 0 at a critical point, the function has a local minimum at that point.

If f’'(x) < 0 at a critical point, the function has a local maximum at that point.

If f’'(x) = 0, the test is inconclusive.

First Derivative Test: Alternatively, examine the sign of the first derivative before and after each critical point.

If f’(x) changes from positive to negative at a critical point, the function has a local maximum there.

If f’(x) changes from negative to positive at a critical point, the function has a local minimum there.

2.0Solved Examples on Rate of Change of Quantities

Example 1: A stone is dropped into a calm lake, creating circular waves that expand at a speed of 4 cm per second. At the moment when the radius of the circular wave reaches 10 cm, how quickly is the enclosed area increasing?

Solution:

The area A of a circle with radius r is given by $A = π r^{2}$ .To find the rate of change of the area A with respect to time t, we differentiate:

$\frac{d A}{d t} = \frac{d}{d t} (π r^{2})$

$\frac{d A}{d t} = π \frac{d}{d t} (r^{2})$

$\frac{d A}{d t} = π (2 r \frac{d r}{d t})$

$\frac{d A}{d t} = 2 π r \frac{d r}{d t}$

Given that:

$\frac{d r}{d t} = 4 cm / s$

Therefore:

$\frac{d A}{d t} = 2 π (10) (4) = 80 π$

Thus, the enclosed area is increasing at a rate of $80 π cm^{2} / s$ when r = 10 cm.

Example 2: A balloon is being inflated and its volume is increasing at a constant rate of 5 cubic centimeters per second. At what rate is the radius of the balloon expanding when the radius is 6 cm?

Solution:

The volume V of a sphere with radius r is given by $V = \frac{4}{3} π r^{3}$ . To find the rate of change of the radius r with respect to time t, we differentiate the volume formula with respect to time:

$\frac{d V}{d t} = \frac{d}{d t} (\frac{4}{3} π r^{3})$

$\frac{d V}{d t} = \frac{4}{3} π \cdot 3 r^{2} \frac{d r}{d t}$

$\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$

Given that:

$\frac{d V}{d t} = 5 cm^{3} / s$

We need to find $\frac{d r}{d t}$ when r = 6 cm. Substituting the known values into the differentiated equation:

$5 = 4 π (6)^{2} \frac{d r}{d t}$

$5 = 144 π \frac{d r}{d t}$

$\frac{d r}{d t} = \frac{5}{144 π}$

Therefore, the radius is increasing at a rate of $\frac{5}{144 π} cm / s$ when the radius is 6 cm.

Example 3: Consider the function $f (x) = x^{3} - 3 x^{2} + 4.$

Solution:

Let $f (x) = x^{3} - 3 x^{2} + 4$

Step 1: Find the first derivative: f’(x) = 3x² – 6x

Step 2: Find critical points by setting the first derivative to zero:

⇒ 3x² – 6x = 0

⇒ 3x (x – 2) = 0

⇒ x = 0 or x = 2

Step 3: Use the second derivative test: f’'(x) = 6x – 6

⇒ At x = 0: f’'(0) = 6(0) – 6 = –6 (negative, so local maximum)

⇒ At x = 2: f’'(2) = 6(2) – 6 = 6 (positive, so local minimum)

Therefore, x = 0 is a local maximum and x = 2 is a local minimum.

Example 4: Consider the function $g (x) = x^{4} - 4 x^{3} + 6 x^{2} .$

Solution:

Let $g (x) = x^{4} - 4 x^{3} + 6 x^{2}$

Step 1: Find the first derivative: $g^{'} (x) = 4 x^{3} - 12 x^{2} + 12 x$

Step 2: Find critical points by setting the first derivative to zero:

$\Rightarrow 4 x (x^{2} - 3 x + 3) = 0$

$\Rightarrow x = 0 or solve x^{2} - 3 x + 3 = 0$

Step 3: Use the second derivative test: $g^{''} (x) = 12 x^{2} - 24 x + 12$

⇒ At x = 0: $g^{''} (0) = 12 (0)^{2} - 24 (0) + 12 = 12$ (positive, so local minimum)

Therefore, x = 0 is a local minimum. The quadratic has no real roots, so there are no other critical points.

Example 5: Show that the function $f (x) = 3 x^{2} - 6 x + 4$ is increasing on the set of real numbers $R$ .

Solution:

To determine if the function $f (x) = 3 x^{2} - 6 x + 4$ is increasing on $R$ , we need to find its first derivative and analyze its sign.

Find the first derivative: $f^{'} (x) = \frac{d}{d x} (3 x^{2} - 6 x + 4) = 6 x - 6$
Simplify the derivative: $f ’ (x) = 6 (x -1)$
Analyze the sign of the derivative: Since 6(x – 1) is always positive for all $x \in R$ , the function f(x) is increasing on $R$ .

Therefore, the function $f (x) = 3 x^{2} - 6 x + 4$ is increasing on the set of real numbers $R$ .

Example 6: Determine the intervals on which the function $f (x) = sin x - cos x$ is increasing or decreasing for $0 \leq x \leq 2 π$

Solution:

To find the intervals where the function $f (x) = sin x - cos x$ is increasing or decreasing, we need to analyze its first derivative.

Find the first derivative: $f^{'} (x) = cos x + sin x$
Determine where f’(x) is positive or negative: $f^{'} (x) > 0 \Rightarrow cos x + sin x > 0$

$f^{'} (x) < 0 \Rightarrow cos x + sin x < 0$

Solve the inequality: The points $\frac{π}{4}$ and $\frac{5 π}{4}$ divide the interval [0,2 \pi] into three subintervals:

$[0, \frac{π}{4}], [\frac{π}{4}, \frac{5 π}{4}]$ and $[\frac{5 π}{4}, 2 π]$

For $0 < x < \frac{π}{4}, cos x + sin x > 0$ , so f(x) is increasing.
For $\frac{π}{4} < x < \frac{5 π}{4}, cos x + sin x < 0$ , so f(x) is decreasing.
For $\frac{5 π}{4} < x < 2 π, cos x + sin x > 0$ , so f(x) is increasing.

Therefore, the function $f (x) = sin x - cos x$ is increasing on $[0, \frac{π}{4}]$ and $[\frac{5 π}{4}, 2 π]$ and decreasing on $[\frac{π}{4}, \frac{5 π}{4}]$ .

3.0Practice Questions on Applications of Derivatives

A spherical balloon is being inflated, causing its volume to increase at a rate of 10 cubic centimeters per second. Determine how quickly the radius of the balloon is growing when the radius measures 5 cm.
The radius of a circular oil spill is increasing at a rate of 0.1 meters per minute. How fast is the area of the oil spill increasing when the radius is 20 meters?
A conical tank is being filled with water at a speed of 5 cubic meters per minute. The tank has a height of 10 meters and a base radius of 3 meters. How fast is the water level rising when the water is 6 meters deep?

4.0Sample Questions on Applications of Derivatives

What is meant by the rate of change of a function?

Ans: The rate of change of a function y = f(x) w.r.t x is the derivative $\frac{d y}{d x}$ . It measures how y changes as x varies. For example, in physics, if s(t) represents displacement as a function of time t, then $\frac{d s}{d t}$ gives the velocity.

How do you find the equation of a tangent line using derivatives?

Ans: To find the tangent line to a curve y = f(x) at a point (x₁, y₁):

Find the derivative f'(x), which gives the slope of the tangent at any point x.
Evaluate f'(x) at x₁ to get the slope m.
Use the point-slope form to express the equation of a line: $y - y_{1} = m (x - x_{1})$

1.0Learning Outcomes

After completing this chapter, you will be able to:

Interpret the geometric meaning of derivatives.
Determine intervals where functions are increasing or decreasing.
Find local maxima and minima of functions.
Solve optimization problems involving maxima and minima.
Apply derivatives to calculate rates of change.
Analyze tangents and normals to curves.
Solve application-based problems for JEE and CBSE exams.

2.05-Minute Quick Revision

Must-Revise Concepts

Increasing and Decreasing Functions
Monotonicity
First Derivative Test
Second Derivative Test
Local Maxima and Minima
Absolute Maxima and Minima
Tangents and Normals
Rate of Change
Approximation using Differentials
Optimization Problems

Important Formula Recall

(f'(x)>0) ⇒ Increasing Function
(f'(x)<0) ⇒ Decreasing Function
Critical Point: (f'(x)=0) or undefined
Maximum/Minimum Conditions
Slope of Tangent = (f'(x))
Slope of Normal = (-\frac{1}{f'(x)})
Linear Approximation: dy=f'(x),dx

Important Properties / Shortcut Tips

Always identify the critical points before applying derivative tests.
Use the first derivative test to determine increasing and decreasing intervals.
Use the second derivative test only after finding stationary points.
Carefully interpret the question before choosing the optimization variable.
Draw rough graphs whenever possible to visualize maxima, minima, and monotonicity.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Increasing & Decreasing Functions	High	Easy–Moderate
First Derivative Test	Very High	Moderate
Second Derivative Test	Very High	Moderate
Maxima & Minima	Very High	Moderate–High
Tangents & Normals	High	Moderate
Rate of Change	High	Easy–Moderate
Optimization Problems	Very High	High
Approximation using Differentials	Moderate	Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Application of Derivatives in CBSE Board, JEE Main, and JEE Advanced examinations.

5.0Common Mistakes & JEE Tips

Students often lose marks in Application of Derivatives due to incorrect interpretation of derivative tests and optimization conditions.

Common Mistake	Correct Approach
Ignoring critical points	Find all points where (f'(x)=0) or undefined before testing.
Using only the second derivative test	Apply the first derivative test when the second derivative is inconclusive.
Incorrect sign analysis	Check the sign of (f'(x)) carefully in each interval.
Wrong slope of the normal	Use (m=-\frac{1}{f'(x)}), not (f'(x)).
Choosing the wrong optimization variable	Define variables and constraints before differentiating.
Ignoring the domain of the function	Verify that the obtained solution lies within the valid domain.
Algebraic errors after differentiation	Simplify expressions carefully before solving.
Forgetting endpoint values	Compare endpoint values whenever absolute maxima or minima are required.

6.0ALLEN Faculty Tips

Master Continuity and Differentiability before starting this chapter.
Practice sign chart questions for increasing and decreasing functions.
Learn both the first and second derivative tests thoroughly.
Draw rough graphs to understand maxima, minima, and monotonicity.
Practice optimization questions from different real-life scenarios.
Revise tangent and normal formulas regularly.
Solve Previous Years' Questions (PYQs) to improve speed and accuracy.

7.0PYQ Trend Analysis

Application of Derivatives is one of the highest-weightage chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are frequently asked on monotonicity, maxima and minima, tangents, normals, rate of change, and optimization.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Increasing & Decreasing Functions	1–2	Moderate
Maxima & Minima	2–3	Moderate to Difficult
First & Second Derivative Tests	1–2	Moderate
Tangents & Normals	1	Moderate
Rate of Change	1	Easy to Moderate
Optimization Problems	2–3	Moderate to Difficult

8.0Most Common Question Types

Finding intervals where a function is increasing or decreasing.
Determining local and absolute maxima or minima.
Applying first and second derivative tests.
Finding equations of tangents and normals.
Solving optimization problems involving area, volume, and cost.
Calculating rates of change in practical situations.
Identifying stationary points and points of inflection.
Solving graph-based and application-oriented questions.

9.0Smart Revision Strategy

Follow this revision sequence to master the chapter efficiently:

Revise derivative formulas from the previous chapter.
Practice increasing and decreasing function problems.
Learn the first and second derivative tests.
Solve maxima and minima questions.
Practice tangent, normal, and rate of change problems.
Revise optimization techniques with different examples.
Solve Previous Years' Questions (PYQs).
Finish with a timed chapter-wise mock test to improve accuracy.

Class 12 Maths Chapter 6: Application of Derivatives

1.0Chapter Snapshot

2.0Chapter 6 Smart Study Material by ALLEN Experts: Download PDFs

Test your Knowledge

question 1 of 5

The derivative of a function at a point represents the:

Frequently Asked Questions

What are the applications of derivatives in real life?

Why are maxima and minima important in Application of Derivatives?

What is the difference between a tangent and a normal?

Which concepts require the most practice in this chapter?

Why is Application of Derivatives important for JEE Main and JEE Advanced?

Join ALLEN!

Class 12 Maths Chapter 6: Application of Derivatives

1.0Chapter Snapshot

2.0Chapter 6 Smart Study Material by ALLEN Experts: Download PDFs

Test your Knowledge

question 1 of 5

The derivative of a function at a point represents the:

Applications of Derivatives

1.0Rate of Change of Quantities

2.0Increasing and Decreasing Functions

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

Frequently Asked Questions

What are the applications of derivatives in real life?

Why are maxima and minima important in Application of Derivatives?

What is the difference between a tangent and a normal?

Which concepts require the most practice in this chapter?

Why is Application of Derivatives important for JEE Main and JEE Advanced?

Join ALLEN!

Applications of Derivatives

1.0Rate of Change of Quantities

2.0Increasing and Decreasing Functions

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy