RD Sharma Solutions for Class 11 Maths Chapter 30 Derivatives

RD Sharma Solutions Class 11 Maths Chapter 30 – Download Free PDF Updated Session for 2023-24

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives are provided here for students to score good marks in the final exam. Chapter 30 of Class 11 RD Sharma Solutions has explanatory answers to exercise-wise problems in a comprehensive manner. This chapter provides students with the knowledge of derivatives at a point. The PDF of RD Sharma Solutions is available, both online and offline, to help students ace the examination.

Chapter 30 – Derivatives contains five exercises, and RD Sharma Solutions provide descriptive answers to the questions present in each exercise. Students who aim to perform well in the final exam are advised to solve problems using the solutions PDF. The main aim of creating solutions is to help students clear their doubts instantly and improve conceptual knowledge. RD Sharma Class 11 Maths Solutions free PDF links are available below. Now, let us have a look at the concepts discussed in this chapter.

  • Derivative at a point
    • Physical interpretation of derivative at a point
    • Geometrical interpretation of derivative at a point
  • Derivative of a function
    • Derivative as a rate measurer
  • Differentiation from first principles
  • Fundamental rules for differentiation
    • Product rule for differentiation
    • Quotient rule for differentiation

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives

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Access answers to RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives

EXERCISE 30.1 PAGE NO: 30.3

1. Find the derivative of f(x) = 3x at x = 2

Solution:

Given:

f(x) = 3x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 1

2. Find the derivative of f(x) = x2 – 2 at x = 10

Solution:

Given:

f(x) = x2 – 2

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 2

= 0 + 20 = 20

Hence,

Derivative of f(x) = x2 – 2 at x = 10 is 20

3. Find the derivative of f(x) = 99x at x = 100.

Solution:

Given:

f(x) = 99x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 3

4. Find the derivative of f(x) = x at x = 1

Solution:

Given:

f(x) = x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 4

5. Find the derivative of f(x) = cos x at x = 0

Solution:

Given:

f(x) = cos x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 5

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 6

6. Find the derivative of f(x) = tan x at x = 0

Solution:

Given:

f(x) = tan x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 7

7. Find the derivatives of the following functions at the indicated points:
(i) sin x at x = π/2
(ii) x at x = 1

(iii) 2 cos x at x = π/2

(iv) sin 2xat x = π/2

Solution:

(i) sin x at x = π/2

Given:

f (x) = sin x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 8

[Since it is of indeterminate form, let us try to evaluate the limit.]

We know that 1 – cos x = 2 sin2(x/2)

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 9

(ii) x at x = 1

Given:

f (x) = x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 10

(iii) 2 cos x at x = π/2

Given:

f (x) = 2 cos x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 11

(iv) sin 2xat x = π/2

Solution:

Given:

f (x) = sin 2x

By using the derivative formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 12

[Since it is of indeterminate form, we shall apply the sandwich theorem to evaluate the limit.]

Now, multiply the numerator and denominator by 2, and we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 13


EXERCISE 30.2 PAGE NO: 30.25

1. Differentiate each of the following from the first principles:
(i) 2/x
(ii) 1/√x

(iii) 1/x3

(iv) [x2 + 1]/ x

(v) [x2 – 1] / x

Solution:

(i) 2/x

Given:

f (x) = 2/x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 14

∴ Derivative of f(x) = 2/x is -2x-2

(ii) 1/√x

Given:

f (x) = 1/√x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 15

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 16

∴ Derivative of f(x) = 1/√x is -1/2 x-3/2

(iii) 1/x3

Given:

f (x) = 1/x3

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 17

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 18

∴ Derivative of f(x) = 1/x3 is -3x-4

(iv) [x2 + 1]/ x

Given:

f (x) = [x2 + 1]/ x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 19

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 20

= 1 – 1/x2

∴ Derivative of f(x) = 1 – 1/x2

(v) [x2 – 1] / x

Given:

f (x) = [x2 – 1]/ x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 21

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 22

2. Differentiate each of the following from the first principles:

(i) e-x

(ii) e3x

(iii) eax+b

Solution:

(i) e-x

Given:

f (x) = e-x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 23

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 24

(ii) e3x

Given:

f (x) = e3x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 25

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 26

(iii) eax+b

Given:

f (x) = eax+b

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 27

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 28

3. Differentiate each of the following from the first principles:

(i) √(sin 2x)

(ii) sin x/x

Solution:

(i) √(sin 2x)

Given:

f (x) = √(sin 2x)

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 29

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 30

(ii) sin x/x

Given:

f (x) = sin x/x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 31

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 32

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 33

4. Differentiate the following from the first principles:

(i) tan2 x

(ii) tan (2x + 1)

Solution:

(i) tan2 x

Given:

f (x) = tan2 x

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 34

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 35

(ii) tan (2x + 1)

Given:

f (x) = tan (2x + 1)

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 36

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 37

5. Differentiate the following from the first principles:

(i) sin √2x

(ii) cos √x

Solution:

(i) sin √2x

Given:

f (x) = sin √2x

f (x + h) = sin √2(x+h)

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 38

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 39

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 40

(ii) cos √x

Given:

f (x) = cos √x

f (x + h) = cos √(x+h)

By using the formula,

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 41

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 42

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 43


EXERCISE 30.3 PAGE NO: 30.33

Differentiate the following with respect to x:

1. x4 – 2sin x + 3 cos x

Solution:

Given:

f (x) = x4 – 2sin x + 3 cos x

Differentiate on both sides with respect to x, and we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 44

2. 3x + x3 + 33

Solution:

Given:

f (x) = 3x + x3 + 33

Differentiate on both sides with respect to x, and we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 45

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 46

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 47

Solution:

Given:

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 48

Differentiate on both sides with respect to x, and we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 49

4. ex log a + ea log x + ea log a

Solution:

Given:

f (x) = ex log a + ea log x + ea log a

We know that,

elog f(x) = f(x)

So,

f(x) = ax + xa + aa

Differentiate on both sides with respect to x, and we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 50

5. (2x2 + 1) (3x + 2)

Solution:

Given:

f (x) = (2x2 + 1) (3x + 2)

= 6x3 + 4x2 + 3x + 2

Differentiate on both sides with respect to x, and we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 51


EXERCISE 30.4 PAGE NO: 30.39

Differentiate the following functions with respect to x:

1. x3 sin x

Solution:

Let us consider y = x3 sin x

We need to find dy/dx

We know that y is a product of two functions, say u and v, where,

u = x3 and v = sin x

∴ y = uv

Now let us apply the product rule of differentiation.

By using the product rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 52

2. x3 ex

Solution:

Let us consider y = x3 ex

We need to find dy/dx

We know that y is a product of two functions, say u and v, where,

u = x3 and v = ex

∴ y = uv

Now let us apply the product rule of differentiation.

By using the product rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 53

3. x2 ex log x

Solution:

Let us consider y = x2 ex log x

We need to find dy/dx

We know that y is a product of two functions, say u and v, where,

u = x2 and v = ex

∴ y = uv

Now let us apply the product rule of differentiation.

By using the product rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 54

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 55

4. xn tan x

Solution:

Let us consider y = xn tan x

We need to find dy/dx

We know that y is a product of two functions, say u and v, where,

u = xn and v = tan x

∴ y = uv

Now let us apply the product rule of differentiation.

By using the product rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 56

5. xn loga x

Solution:

Let us consider y = xn loga x

We need to find dy/dx

We know that y is a product of two functions, say u and v, where,

u = xn and v = loga x

∴ y = uv

Now let us apply the product rule of differentiation.

By using the product rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 57


EXERCISE 30.5 PAGE NO: 30.44

Differentiate the following functions with respect to x:

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 58

Solution:

Let us consider

y =
RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 59

We need to find dy/dx

We know that y is a fraction of two functions, say u and v, where,

u = x2 + 1 and v = x + 1

∴ y = u/v

Now let us apply the quotient rule of differentiation.

By using the quotient rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 60

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 61

Solution:

Let us consider

y =
RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 62

We need to find dy/dx

We know that y is a fraction of two functions, say u and v, where,

u = 2x – 1 and v = x2 + 1

∴ y = u/v

Now let us apply the quotient rule of differentiation.

By using the quotient rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 63

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 64

Solution:

Let us consider

y =
RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 65

We need to find dy/dx

We know that y is a fraction of two functions, say u and v, where,

u = x + ex and v = 1 + log x

∴ y = u/v

Now let us apply the quotient rule of differentiation.

By using the quotient rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 66

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 67

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 68

Solution:

Let us consider

y =
RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 69

We need to find dy/dx

We know that y is a fraction of two functions, say u and v, where,

u = ex – tan x and v = cot x – xn

∴ y = u/v

Now let us apply the quotient rule of differentiation.

By using the quotient rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 70

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 71

Solution:

Let us consider

y =
RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 72

We need to find dy/dx

We know that y is a fraction of two functions, say u and v, where,

u = ax2 + bx + c and v = px2 + qx + r

∴ y = u/v

Now let us apply the quotient rule of differentiation.

By using the quotient rule, we get

RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives - image 73

Also, access exercises of RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives

Exercise 30.1 Solutions

Exercise 30.2 Solutions

Exercise 30.3 Solutions

Exercise 30.4 Solutions

Exercise 30.5 Solutions

Frequently Asked Questions on RD Sharma Solutions for Class 11 Maths Chapter 30

Q1

Why should students practise RD Sharma Solutions for Class 11 Maths Chapter 30?

Practising RD Sharma Solutions for Class 11 Maths Chapter 30 provides students with an idea about the important questions that would be asked in the final exam. Regular use of these solutions helps students to analyse the weak areas they are lagging behind in, which need more focus. The solutions explain all the covered topics included in the textbook comprehensively, as per the current CBSE board. A team of expert faculty at BYJU’S designed the solutions with the aim of providing students with the best resource for in-depth learning of concepts.
Q2

Can RD Sharma Solutions for Class 11 Maths Chapter 30 be viewed only online?

No, RD Sharma Solutions for Class 11 Maths Chapter 30 can be viewed both online and offline, to obtain detailed knowledge of concepts anytime. By regular use of these solutions, students can understand numerous methods, tricks and tips to solve any type of problem efficiently. The solutions PDF are solved by a highly experienced faculty team at BYJU’S, prescribed by the CBSE guidelines and exam patterns. Students can download the solutions PDFs for effective exam preparation as per their needs.
Q3

Is it important to learn all the questions provided in RD Sharma Solutions for Class 11 Maths Chapter 30?

Yes, students must learn all the questions provided in RD Sharma Solutions for Class 11 Maths Chapter 30. This is because these questions may appear in the final exam and help to secure high marks. The solutions created by the experts are of high quality, and the free download links can be used by the students to access them anytime. Practising these questions on a daily basis helps students to write precise and elaborate answers for their upcoming exams more confidently. This also helps them to improve their problem-solving and time-management skills, which are vital for the exam.
Q4

How do RD Sharma Solutions for Class 11 Maths Chapter 30 boost exam preparation among CBSE students?

RD Sharma Solutions for Class 11 Maths Chapter 30 explain concepts in a precise manner with the goal of providing the best study sources for effective exam preparation among CBSE students. Practising RD Sharma Solutions on a regular basis helps students clear their confusion on the spot and gain skills in Mathematics. Following these solutions while solving textbook problems definitely improves better academic performance among CBSE students. For more conceptual knowledge, students can access the solutions in PDF format from BYJU’S whenever required.
Q5

What are the concepts students learn in RD Sharma Solutions for Class 11 Maths Chapter 30?

The concepts students learn in RD Sharma Solutions for Class 11 Maths Chapter 30 are listed below.

  • Derivative at a point
    • Physical interpretation of derivative at a point
    • Geometrical interpretation of derivative at a point
  • Derivative of a function
    • Derivative as a rate measurer
  • Differentiation from first principles
  • Fundamental rules for differentiation
    • Product rule for differentiation
    • Quotient rule for differentiation

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