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# RD Sharma Class 12 Solutions Chapter 11 - Differentiation (Ex 11.5) Exercise 11.5

Last updated date: 23rd May 2024
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## RD Sharma Class 12 Solutions Chapter 11 - Differentiation (Ex 11.5) Exercise 11.5 - Free PDF

Download of RD Sharma Class 12 Solutions Chapter 11 - Differentiation Exercise 11.5 solved by Expert Mathematics Teachers on Vedantu. All Chapter 11 - Differentiation Ex 11.5 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise the complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other engineering entrance exams.

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## Class 12 Solutions Chapter 11 - Differentiation

The chapter

‘Differentiation’ is Chapter 11 of RD Sharma Class 12 Maths to help the students to learn about differentiation and how to find derivatives of simple functions. The two important parts of calculus are differentiation and calculus. The process of finding the derivative, or rate of change, of a function, is known as Differentiation. Differentiation is the process that can be carried out by purely algebraic manipulations, using three basic derivatives and four rules of operations.

Differentiation can be applied to measure the function per unit change in the independent variable. An example of differentiation is that the speed is calculated as the rate of change of distance concerning time a very basic derivatives (D) are-

1. for algebraic functions, D (xn)= nxn-1, in which n is any real number;

2. for trigonometric functions, D(sin x)= cos x; and

3. for exponential functions, D(ex)= ex

### Formulas Used in Differentiation

#### Basic Differentiation Formulas

Let us consider f(x) is a function and f’x is a derivative of the function

 value of f(x) is Value of f’x is tan(x) sec2x cos(x) -sin x sin(x) cos x ln(x) 1/x ex ex xn nxn-1 k - where k is a constant 0

### Rules for Differentiation

There are 4 rules of Differentiation-

#### 1. Sum and Difference Rule

The derivative of a sum is equal to the sum of the derivatives

$\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

The derivative of a difference is equal to the difference of the derivatives

$\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}[f(x)] - \frac{d}{dx}[g(x)]$

#### 2. Product Rule

If f(x).g(x) is a function, then the derivative of the function is

$\frac{d}{dx}[f(x).g(x)] = {f}'x.g(x) + f(x).{g}'(x)$

#### 3. Quotient Rule

If f(x)=u(x)/v(x) is a function, then the derivative of the function is

${f}'x={[\frac{u(x)}{v(x)}]}'=\frac{v(x)\times {u}'(x)-u(x)\times {v}'(x)}{[v(x)]^{2}}$

#### 4. Chain Rule

If y=f(x)=g(u) and if u=h(x), then the chain rule for differentiation will be

$\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$

### Advantages of RD Sharma Maths for Class 11

If the students are preparing for competitive exams they should definitely go for RD Sharma Maths for Class 11. RD Sharma Maths Class 11 provides a clearer understanding of each and every topic. Since the main goal is to maintain quality while studying, not quantity there it is best to refer to RD Sharma books in which all topics are explained in detail and systematic way.

There are Many Other Advantages of RD Sharma Maths for Class 11 such as-

• One of the main advantages of RD Sharma is that there are various types of questions available. It has basic level questions for mediocre students as well as for the students who want to test their critical thinking, they have Higher Order Thinking Skill questions.

• Another advantage is that all the questions are divided into different sub-types so that the students are aware of all the types of questions that can be formed out of one question and can be asked in an exam

• All the topics are explained in detail which gives the student a clear picture of the concept and helps them to study the concept to its core.

## FAQs on RD Sharma Class 12 Solutions Chapter 11 - Differentiation (Ex 11.5) Exercise 11.5

1. Where can I find revision notes for Class 12 Differentiation?

For Class 12 Maths Differentiation, Vedantu provided Revision Notes. The revision notes are designed by various subject experts and follow the latest exam pattern. These notes will help the students to learn all the various types of differential equations and an in-depth study about these equations. Relevant diagrams are also provided in the notes to help students to have a better understanding of the concept. Going through these notes thoroughly will help the students to even attempt the complicated exams easily and with much more confidence in the exam.

2. What are the real-life applications of all the four rules of differentiation- chain rule, sum and difference rule, product rule and quotient rule?
Chain Rule

In Engineering, the chain rule has broad applications-

• To determine the increase and decrease in a function.

• To calculate between two moving objects, the rate of change of distance.

• It helps us to deduce the rate of change in the real world.

Sum and Difference Rule
• To determine the long distance between two points such as the distance between the sun and the earth.

• To determine long heights such as the height of a mountain.

• To determine how long the drug will take to enter your body.

Product Rule
• To generate accounting reports in business

• To determine the effect of price change in revenue

Quotient Rule

The quotient rule and the other rules can be used interchangeably but one example could be, given the volume of pool water and the rate of water flowing into the pool if you were filling a pool and you want to see how much the water has increase you can use the quotient rule.

3. What is the difference between a derivative and a differential?

The differential is one of the fundamental divisions of calculus. The differential is a subfield of calculus that deals with infinitesimal changes in varying quantities. Differential equations consist of unknown functions and their derivatives while a derivative of a function is a new function also called the rate function.

The differential is the function that represents the relationship between dependent and independent variables while the derivative is the function where the relationship between the variables is known.

4. Are Vedantu’s RD Sharma Class 12 Solutions helpful?

Vedantu designs all its solutions from the subject experts themselves and all the solutions are conceptually and methodologically. Most of the students who go for competitive exams prefer RD Sharma Class 12 Maths which has higher level questions as well and sometimes it becomes difficult for the students to solve.

Vedantu’s solutions are designed in such a way that they not only just answer the questions but help the students to make the conceptual understanding in a better manner.

5. How to study differential equations?

The most important part while studying differential equations is practice. Once the student learns all the differential equations it becomes very easy for the student to solve the questions. If the student faces any problem while solving any question or understanding any concept. Vedantu provides everything, be it revision notes or solved questions.