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NCERT Solutions for Class 11 Maths Chapter 12 Exercise 12.2 Limits and Derivatives - PDF Download

JEE Mains & Advanced

NCERT Solutions for class 11 maths chapter 12 exercise 12.2 is now focusing on derivatives and related concepts. This exercise focuses exclusively on teaching students how to identify the derivatives of various functions, including algebra of derivatives of functions, polynomial functions and trigonometric functions. Students will get clear and precise explanations of all these topics by eSaral’s expert faculty.

Class 11 maths chapter 12 exercise 12.2 NCERT solutions consists of 11 questions that are related to derivatives and their algebras. It is important to remember that when solving the questions of this exercise, it is always necessary to verify the result by using the basic techniques provided in NCERT solutions class 11 maths chapter 12 ex 12.2 Limits and Derivatives. In order to help students in achieving high scores in final exams, eSaral’s subject experts have provided answers to all the questions in chapter 12 ex 12.2 class 11 maths. These NCERT solutions for ex 12.2 class 11 maths ch 12 are also available in PDF format at eSaral which can be downloaded for free. You can download the free PDF from the link given below.

Topics Covered in Exercise 12.2 Class 11 Mathematics Questions

Ex 12.2 class 11 maths chapter 12 is completely based on the topics like derivatives, algebra of derivative of functions and derivative of polynomials and trigonometric functions that are explained in detail below.

1.

Derivatives

2.

Algebra of derivative of functions

3.

Derivative of polynomials and trigonometric functions

  1. Derivatives - Derivative of a function at a given point in its domain of definition.

Definition 1 - Suppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by 

                            $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$

provided this limit exists. Derivative of f (x) at a is denoted by f′(a). provided this limit exists. Derivative of f (x) at a is denoted by f′(a).

Definition 2 - Suppose f is a real valued function, the function defined by

                            $\lim _{h \rightarrow 0} \frac{f(x+h)-F(x)}{h}$

wherever the limit exists is defined to be the derivative of f at x and is denoted by f′(x). This definition of derivative is also called the first principle of derivative.

Thus, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

Clearly the domain of definition of f′ (x) is wherever the above limit exists. There are different notations for derivatives of a function. Sometimes f′(x) is denoted by $\frac{d}{d x}$ (f(x)) or if y = f(x), it is denoted by $\frac{d y}{d x}$ This is referred to as derivative of f(x) or y with respect to x. It is also denoted by D (f (x) ). Further, derivative of f at x = a is also denoted by 

                          $\frac{d}{d x} f(x) \mid a$ or $\frac{d f}{d x} \mid a$ or even $\left(\frac{d f}{d x}\right)_{x=a}$

  1. Algebra of derivative of functions - The rules for derivatives are expected to be approximately the same as those of limits. The rules of derivative are defined in the following theorem.

Theorem 5: Let f and g be two functions such that their derivatives are defined in a common domain. Then

(i) Derivative of sum of two functions is sum of the derivatives of the functions.

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

(ii) Derivative of difference of two functions is difference of the derivatives of the functions.

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

(iii) Derivative of product of two functions is given by the following product rule.               

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

(iv) Derivative of quotient of two functions is given by the following quotient rule (whenever the denominator is non–zero).

                      $\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right)=\frac{\frac{d}{d x} f(x) \cdot g(x)-f(x) \frac{d}{d x} g(x)}{(g(x))^2}$

Theorem 6: Derivative of f(x)=xn is nxn-1 for any positive integer n. 

  1. Derivative of polynomials and trigonometric functions - Here, you will learn some theorems related to derivative of polynomials and trigonometric functions.

Theorem 7: Let f(x)= anxn+an-1xn-1+....+a1x+a0 be a polynomial function, where ais are all real numbers and an≠ 0 Then, the derivative function is given by

           $\frac{d f(x)}{d x}=n a_n x^{n-1}+(n-1) a_{n-1} x^{x-2}+\ldots+2 a_2 x+a_1$.

Tips for Solving Exercise 12.2 Class 11 Chapter 12 Limits and Derivatives

You can find here some useful tips and tricks to solve questions of ex 12.2 class 11 maths chapter 12 Limits and Derivatives.

  1. NCERT solutions class 11 maths chapter 12 ex 12.2 Limits and Derivatives have specific formulas that will be extensively used in ex 12.2 to solve questions.

  2. NCERT solutions class 11 maths chapter 12 exercise 12.2 also includes the proof of several theorems based on the fundamental algebraic operations of derivatives that will help you to solve complex problems of ex 12.2 class 11 maths chapter 12.

  3. You must read out all the topics and concepts associated with ex 12.2 to be well versed with the questions.

Importance of Solving Ex 12.2 Class 11 Maths Chapter 12 Limits and Derivatives

Ex 12.2 class 11 maths solutions has questions which are based on derivative and its properties. Here, we have provided some of the essential benefits of solving ex 12.2 class 11 maths chapter 12 limits and derivatives.

  1. Ex 12.2 class 11 maths chapter 12 provides a comprehensive description of derivative and its associated concepts in the NCERT solutions, which is presented in an easy-to-understand, stepwise format to help you understand and apply problem solving techniques.

  2. Using a formula chart as a starting point and practicing it regularly will help students understand the concepts better.

  3. Class 11 maths chapter 12 exercise 12.2 NCERT solutions are typically available in free PDF, providing an accessible resource for self-study.

  4. These class 11 ex 12.2 solutions can be used by students to cross-check their answers, test their problem-solving skills, and improve their math skills.

Frequently Asked Questions 

Question 1. Why are derivatives important?

Answer 1. A derivative is a numerical expression that expresses a change in the relationship between two independent variables. The formula for a derivative can be applied to determine a line's slope, a curve's slope, and difference between two measurements. That’s why derivatives are so important, and you need to learn them well as well as practice different kinds of questions included in ex 12.2 class 11 maths.

Question 2. Why is it essential to practice questions from NCERT solutions of ex 12.2 class 11 maths chapter 12?

Answer 2. NCERT solutions for ex 12.2 class 11 maths chapter 12 are solved by subject experts of eSaral. You can find here the important concepts, theorems and formulas related to derivative. All the concepts are explained in detailed format by eSaral’s expert teachers. Our subject experts have provided step by step solutions for every question of ex 12.2 chapter 12 that will help you to score good marks in exams.

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