RD Sharma Solutions for Class 11 Chapter 30 - Derivatives

RD Sharma Class 11 Chapter 30 has explanatory answers to exercise wise problems in a comprehensive manner. Chapter 30 provides students with the knowledge of derivatives at a point. The PDF of solutions is made available both chapter wise and exercise wise to help students ace the examination. Students who aim to perform well in the board exams are advised to solve problems using the solutions PDF. The main aim of creating solutions is to help students clear their doubts and improve conceptual knowledge. RD Sharma Class 11 Maths Solutions, free PDF links are available below.

Chapter 30 – Derivatives contains five exercises and the RD Sharma Solutions present in this page provide solutions to the questions present in each exercise. 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.

Download the pdf of RD Sharma Solutions for Class 11 Maths Chapter 30 – Derivatives

 

rd sharma class 11 maths chapter 30 ex 1 1
rd sharma class 11 maths chapter 30 ex 1 2
rd sharma class 11 maths chapter 30 ex 1 3
rd sharma class 11 maths chapter 30 ex 1 4
rd sharma class 11 maths chapter 30 ex 1 5
rd sharma class 11 maths chapter 30 ex 1 6
rd sharma class 11 maths chapter 30 ex 1 7

rd sharma class 11 maths chapter 30 ex 2 01
rd sharma class 11 maths chapter 30 ex 2 02
rd sharma class 11 maths chapter 30 ex 2 03
rd sharma class 11 maths chapter 30 ex 2 04
rd sharma class 11 maths chapter 30 ex 2 05
rd sharma class 11 maths chapter 30 ex 2 06
rd sharma class 11 maths chapter 30 ex 2 07
rd sharma class 11 maths chapter 30 ex 2 08
rd sharma class 11 maths chapter 30 ex 2 09
rd sharma class 11 maths chapter 30 ex 2 10
rd sharma class 11 maths chapter 30 ex 2 11
rd sharma class 11 maths chapter 30 ex 2 12
rd sharma class 11 maths chapter 30 ex 2 13
rd sharma class 11 maths chapter 30 ex 2 14
rd sharma class 11 maths chapter 30 ex 2 15

rd sharma class 11 maths chapter 30 ex 3 1
rd sharma class 11 maths chapter 30 ex 3 2
rd sharma class 11 maths chapter 30 ex 3 3

rd sharma class 11 maths chapter 30 ex 4 1
rd sharma class 11 maths chapter 30 ex 4 2
rd sharma class 11 maths chapter 30 ex 4 3
rd sharma class 11 maths chapter 30 ex 4 4

rd sharma class 11 maths chapter 30 ex 5 1
rd sharma class 11 maths chapter 30 ex 5 2
rd sharma class 11 maths chapter 30 ex 5 3
rd sharma class 11 maths chapter 30 ex 5 4
rd sharma class 11 maths chapter 30 ex 5 5

 

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 sandwich theorem to evaluate the limit.]

Now, multiply numerator and denominator by 2, 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 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 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 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 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 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 the sides with respect to x, 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 the sides with respect to x, 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 the sides with respect to x, 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 the sides with respect to x, 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 the sides with respect to x, 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 product rule of differentiation.

By using 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 product rule of differentiation.

By using 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 product rule of differentiation.

By using 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 product rule of differentiation.

By using 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 product rule of differentiation.

By using 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 quotient rule of differentiation.

By using 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 quotient rule of differentiation.

By using 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 quotient rule of differentiation.

By using 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 quotient rule of differentiation.

By using 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 quotient rule of differentiation.

By using 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

Leave a Comment

Your email address will not be published. Required fields are marked *