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Application of Derivatives Class 12th Notes - Free NCERT Class 12 Maths Chapter 6 Notes - Download PDF

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Application of Derivatives Class 12th Notes - Free NCERT Class 12 Maths Chapter 6 Notes - Download PDF

Edited By Ramraj Saini | Updated on Sep 12, 2022 07:51 PM IST

Introduction: Class 12 Math chapter 6 notes are regarding the Application of Derivatives. Application of derivatives Class 12 notes is about finding the derivatives of the functions. This Class 12 Maths chapter 6 notes contains the following topics: finding the derivatives of the equations, rate of change of quantities, Increasing and decreasing functions, Tangents and normal, Approximations, Maxima and minima, and many more. NCERT Class 12 Math chapter 6 notes also contain theorems and their proofs that are to be remembered for the implementation in problems. NCERT Class 12 Math Chapter 6 contains a detailed explanation of topics, theorems, examples, exercises. By going through the document students can cover all the topics that are in NCERT Notes for Class 12 Math chapter 6 textbook. This also contains examples, exercises, a few interesting points, and most importantly contains FAQ’s that are frequently asked questions by students which can clarify many other students with the same doubt. Every concept that is in CBSE Class 12 Maths chapter 6 notes is explained here in a simple and understanding way that can reach students easily. All these concepts can be downloaded from Class 12 Maths chapter 6 notes pdf download. Class 12 application of derivatives notes, application of derivatives Class 12 notes pdf download. Notes also contain a few of previous year’s question papers.

Also, students can refer,

NCERT Class 12 Chapter 6 Notes

Functions were discussed in chapter 5, now in this chapter, we are going to discuss the applications of those derivatives like a) determining the rate of change, b) finding the equations of tangent and normal to a curve at a point, c) to finding turning points on the graph of a function. We also will be finding increasing and decreasing functions.

Rate of change:

According to CBSE Class 12 chapter 6 notes the rate of change is:

DERIVATIVE:

DEFINITION: The rate of change of distance (S) with respect to time (t) is called the rate of change.

Mathematical Representation: \frac{ds}{dt}\\

Similarly, when one quantity (x) differs with another quantity (y), satisfying y=f(x) then \frac{dy}{dx}\\ or f’(x) represents a rate of change with respect to y at x=?0.

Mathematical Representation: \frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx}\ if\ \frac{dx}{dt}\neq 0

SO, the rate of change of y with x can be calculated using the rate of change of y and x with respect to t.

Increasing and Decreasing Functions:

DEFINITION: Strictly increasing:

function f:X→R is defined on an X⊂R is increasing if f(x)≤f(y) where x<y. If there is inequality that is strict, i.e., f(x)<f(y) where x<y, then f is strictly increasing. Strictly decreasing:

function f:X→R is defined on an X⊂R is decreasing if f(x)≥f(y) where x<y. If there is inequality that is strict, i.e., f(x)>f(y) where x<y, then f is strictly decreasing.

Mathematical equations:

a) increasing if x1 < x2 ⇒ f (x1) < f (x2) for all x1, x2 ∈ (interval)

b) decreasing if x1 < x2 ⇒f(x1) < f(x2) for all x1, x2 ∈ (interval)

c) if f(x) = c for all x ∈ intervals, where c is a constant.

d) decreasing if x1 < x2 ⇒ f (x1) ≥ f (x2) for all x1, x2 ∈ intervals.

e) strictly decreasing if x1 < x2 ⇒ f(x1) > f(x2) for all x1, x2 ∈ interval THEOREM:

Let f which denotes a function be a continuous function on [a, b] and is differentiable on the interval (a, b). Then

(a) f is said to be increasing in interval [a, b] if f ′(x) > 0 where x ∈ (a, b)

(b) f is said to be decreasing in interval [a, b] if f ′(x) < 0 where x ∈ (a, b)

(c) f is a said to be a constant function interval in [a, b] if f ′(x) = 0 where x ∈ (a, b)

Proof: Let x1, x2 ∈ to interval [a, b] where x1 < x2. Then, by Mean Value Theorem that is learned earlier \\ f'(c)=\frac{f(x2)-f(x1)}{(x2-x1)} \Rightarrow\ f(x2)-f(x1) \ = \ f'(c)(x2-x1)\\ \Rightarrow f(x2)-f(x1) > 0

f(x2) > f(x1)

So x1<x2 in interval (a, b).

Hence, f is said to be an increasing function in [a, b].

Similarly, it can be done to decrease functions as well. By simply interchanging the signs.

Graph for the Functions:

1

Tangents and Normals:

We all know that a line passing through points (?0, ?0) having slope m is given by:

y - ?0 = ?(? − ?0)

The slope of the tangent of the curve y = f(x) at point (?0, ?0) is \frac{dy}{dx}\\ at (?0, ?0).

Equation of tangent: y –?0 = f ‘(?0 )(x – ?0) NOTE:

a) If a tangent makes a line with X-axis in a positive direction, then the slope of the tangent is tan ?.

b) If the slope is 0°, that means the lines are parallel to X-axis, then tan ? = 0;

Then the tangent equation is: y=?0

c) If the slope is 90°, that means the lines are parallel to the y-axis, then tan ? = infinity;

Then the tangent equation is x = ?0

NORMAL: Normal is perpendicular to the tangent. So, the slope of normal to the -1/f’(x).

Equation of normal is: (y –?0) f ‘(?0 )+(x – ?0)=0;

APPROXIMATIONS:

A function f: D → R, D is a subset of R (Real numbers), be function and y = f (x).

Then the small increment ∆? ?? ? ??????? ?? ∆? ?? ????? ??

dy = f (x + ∆x) – f (x).

This is called approximation.

a) here we have dx\approx \Delta x

b) dy\approx \frac{dy}{dx} \Delta x

c)dy \approx \Delta y

Maxima and Minima :

If f be a function on an interval I. Then

(a) f will have a maximum value, if there is a point c in the interval in a way that f (c) > f(x), in the interval. Then f(c) is called the maximum value of the function f and that point where the maximum value is found is said to be a point of maximum for function f.

(b) b) f will have a minimum value, if there is a point c in the interval in a way that f (c) < f(x), in the interval. Then f(c) is called the minimum value of the function f and that point where the minimum value is found is said to be a point of minimum for function f.

(c) f(c) is said to be extreme if it is either a maximum or a minimum value. Point c is said to have an extreme value at that extreme point.

First Derivative Test:

f be a function in the interval I, f be continuous along with point c in I, then

a) When f’(x) changes sign from positive to negative when x increases passing through c, that is f’(x)>0 to left of c and f’(x)<0 to right of c, then c is called point of maxima.

b) When f’(x) changes sign from negative to positive when x increases passing through c, that is f’(x)<0 to left of c and f’(x)>0 to right of c, then c is called point of minima.

c) When f’(x) does not change when x increases, then c is neither a point of local maxima nor a point of local minima. It is called the point of inflection.

Second Derivative Test:

F be a function in the interval I, then f is the twice derivable (f’’(x)) at c, then

a) x=c, a point of local maxima only when f’(c) =0 and f’’(c) < 0 then f(C) is called local maximum of the function f.

b) x=c, a point of local minima only when f’(c) =0 and f’’(c) > 0 then f(C) is called a local minimum of the function f.

c) It doesn’t work if both f’(c)=0 and f’’(c)=0;

Absolute Maximum and Minimum Values:

f is continuous in interval I= [a, b]. f has both absolute

maximum and minimum values and f has at least one value in the interval I;

Theorem: f be a continuous function in interval I. f is a differentiable function in Interval I. c be any point

a) f’(c) =0 if f gets its absolute maximum value at point c.

b) f’(c) =0 if f gets its absolute minimum value at point c.

We have a few working rules to find these values:

Step 1: Finding the critical points

Find f’(x) =0 or not differentiable;

Step 2: Take the extreme points in the interval;

Step 3: calculate all the values of f found above in step 1,2;

Step 4: Find maximum and minimum values that are found in step 3. Among those maximum values, the greatest value of f will be the absolute maximum, and the least value will be the absolute minimum value of f.

Significance of NCERT Class 12 Maths Chapter 6 Notes:

NCERT Class 12 Maths chapter 6 notes will be very much helpful for students to score maximum marks in their 12 board exams. In Application of Derivatives Class 12 chapter 6 notes, we have discussed many topics like finding rate of change, maximum and minimum values, tangent and normal, local maximum and minimum values, absolute maximum and maximum values along with their conditions and rules of solving the problems. NCERT Class 12 Mathematics chapter 6 is also very useful to cover major topics of the Class 12 CBSE Mathematics Syllabus.

The CBSE Class 12 Maths chapter 6 will help to understand the theorems, examples along solutions in detail. This pdf also contains previous year’s questions and NCERT TextBook pdf. The next part contains FAQ’s most frequently asked questions along with topic-wise explanations. By referring to the document you can get a complete idea of all the topics of Class 12 chapter 6 Application of Derivatives pdf download.

NCERT Class 12 Notes Chapter Wise.


Go through the above link to learn more examples topic-wise.

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Frequently Asked Question (FAQs)

1. Where do we use the Application of Derivatives class 12 notes in real life?

1) to calculate profit and loss

2) to find distance, speed, etc

3) for checking temperature

2. Who was the person who bought NCERT class 12 maths chapter 6 notes into the picture for the first time?

Isaac Newton and Gottfried Wilhelm Leibniz were the persons to bring derivatives into picture.

3. What is the reason for the second derivative in NCERT class 12th Math chapter 6 notes?

To find the derivative of the first derivative we use the second derivative.

4. What is used to calculate derivatives according to NCERT notes for class 12 maths chapter 6?

To find functions and to create infinite series. To find whether a function is increasing and decreasing, to find maxima and minima values. 

5. What are the suggested textbooks for application of derivatives?

First basic and best book is NCERT class12 chapter 6 textbook.


Link for NCERT class12 chapter 6 textbook is given above before the significance

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Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook. 

5 Jobs Available
Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

3 Jobs Available
Travel Journalist

The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.

2 Jobs Available
Videographer
2 Jobs Available
SEO Analyst

An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns. 

2 Jobs Available
Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

5 Jobs Available
QA Manager
4 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
Production Manager
3 Jobs Available
Reliability Engineer

Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment. 

2 Jobs Available
Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available
Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
QA Manager
4 Jobs Available
Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

4 Jobs Available
AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

4 Jobs Available
Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
ITSM Manager
3 Jobs Available
Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

3 Jobs Available
Big Data Analytics Engineer

Big Data Analytics Engineer Job Description: A Big Data Analytics Engineer is responsible for collecting data from various sources. He or she has to sort the organised and chaotic data to find out patterns. The role of Big Data Engineer involves converting messy information into useful data that is clean, accurate and actionable. 

2 Jobs Available
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