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# RD Sharma solutions for Class 11 Mathematics chapter 30 - Derivatives

## Chapter 30: Derivatives

Ex. 30.10Ex. 30.20Ex. 30.30Ex. 30.40Ex. 30.50Others

#### Chapter 30: Derivatives Exercise 30.10 solutions [Page 3]

Ex. 30.10 | Q 1 | Page 3

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

Ex. 30.10 | Q 2 | Page 3

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

Ex. 30.10 | Q 3 | Page 3

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

Ex. 30.10 | Q 4 | Page 3

Find the derivative of f (xx at x = 1

Ex. 30.10 | Q 5 | Page 3

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

Ex. 30.10 | Q 6 | Page 3

Find the derivative of (x) = tan x at x = 0

Ex. 30.10 | Q 7.1 | Page 3

Find the derivative of the following function at the indicated point:

sin x at x =$\frac{\pi}{2}$

Ex. 30.10 | Q 7.2 | Page 3

Find the derivative of the following function at the indicated point:

Ex. 30.10 | Q 7.3 | Page 3

Find the derivative of the following function at the indicated point:

2 cos x at x =$\frac{\pi}{2}$

Ex. 30.10 | Q 7.4 | Page 3

Find the derivative of the following function at the indicated point:

sin 2x at x =$\frac{\pi}{2}$

#### Chapter 30: Derivatives Exercise 30.20 solutions [Pages 25 - 26]

Ex. 30.20 | Q 1.01 | Page 25

$\frac{2}{x}$

Ex. 30.20 | Q 1.02 | Page 25

$\frac{1}{\sqrt{x}}$

Ex. 30.20 | Q 1.03 | Page 25

$\frac{1}{x^3}$

Ex. 30.20 | Q 1.04 | Page 25

$\frac{x^2 + 1}{x}$

Ex. 30.20 | Q 1.05 | Page 25

$\frac{x^2 - 1}{x}$

Ex. 30.20 | Q 1.06 | Page 25

$\frac{x + 1}{x + 2}$

Ex. 30.20 | Q 1.07 | Page 25

$\frac{x + 2}{3x + 5}$

Ex. 30.20 | Q 1.08 | Page 25

k xn

Ex. 30.20 | Q 1.09 | Page 25

$\frac{1}{\sqrt{3 - x}}$

Ex. 30.20 | Q 1.1 | Page 25

x2 + x + 3

Ex. 30.20 | Q 1.11 | Page 25

(x + 2)3

Ex. 30.20 | Q 1.12 | Page 25

(x2 + 1) (x − 5)

Ex. 30.20 | Q 1.13 | Page 25

(x2 + 1) (x − 5)

Ex. 30.20 | Q 1.14 | Page 25

$\sqrt{2 x^2 + 1}$

Ex. 30.20 | Q 1.15 | Page 25

$\frac{2x + 3}{x - 2}$

Ex. 30.20 | Q 2.01 | Page 25

Differentiate each of the following from first principle:

ex

Ex. 30.20 | Q 2.02 | Page 25

Differentiate  of the following from first principle:

e3x

Ex. 30.20 | Q 2.03 | Page 25

Differentiate  of the following from first principle:

eax + b

Ex. 30.20 | Q 2.04 | Page 25

x ex

Ex. 30.20 | Q 2.05 | Page 25

Differentiate  of the following from first principle:

− x

Ex. 30.20 | Q 2.06 | Page 25

Differentiate of the following from first principle:

(−x)−1

Ex. 30.20 | Q 2.07 | Page 25

Differentiate  of the following from first principle:

sin (x + 1)

Ex. 30.20 | Q 2.08 | Page 25

Differentiate  of the following from first principle:

$\cos\left( x - \frac{\pi}{8} \right)$

Ex. 30.20 | Q 2.09 | Page 25

Differentiate  of the following from first principle:

x sin x

Ex. 30.20 | Q 2.1 | Page 25

Differentiate of the following from first principle:

x cos x

Ex. 30.20 | Q 2.11 | Page 25

Differentiate  of the following from first principle:

sin (2x − 3)

Ex. 30.20 | Q 3.01 | Page 26

Differentiate each of the following from first principle:

$\sqrt{\sin 2x}$

Ex. 30.20 | Q 3.02 | Page 26

Differentiate each of the following from first principle:

$\frac{\sin x}{x}$

Ex. 30.20 | Q 3.03 | Page 26

Differentiate each of the following from first principle:

$\frac{\cos x}{x}$

Ex. 30.20 | Q 3.04 | Page 26

Differentiate each of the following from first principle:

x2 sin x

Ex. 30.20 | Q 3.05 | Page 26

Differentiate each of the following from first principle:

$\sqrt{\sin (3x + 1)}$

Ex. 30.20 | Q 3.06 | Page 26

Differentiate each of the following from first principle:

sin x + cos x

Ex. 30.20 | Q 3.07 | Page 26

Differentiate each of the following from first principle:

x2 e

Ex. 30.20 | Q 3.08 | Page 26

Differentiate each of the following from first principle:

$e^{x^2 + 1}$

Ex. 30.20 | Q 3.09 | Page 26

Differentiate each  of the following from first principle:

$e^\sqrt{2x}$

Ex. 30.20 | Q 3.1 | Page 26

Differentiate each of the following from first principle:

$e^\sqrt{ax + b}$

Ex. 30.20 | Q 3.11 | Page 26

Differentiate each of the following from first principle:

$a^\sqrt{x}$

Ex. 30.20 | Q 3.12 | Page 26

Differentiate each of the following from first principle:

$3^{x^2}$

Ex. 30.20 | Q 4.1 | Page 26

tan2

Ex. 30.20 | Q 4.2 | Page 26

tan (2x + 1)

Ex. 30.20 | Q 4.3 | Page 26

tan 2

Ex. 30.20 | Q 4.4 | Page 26

$\sqrt{\tan x}$

Ex. 30.20 | Q 5.1 | Page 26

$\sin \sqrt{2x}$

Ex. 30.20 | Q 5.2 | Page 26

$\cos \sqrt{x}$

Ex. 30.20 | Q 5.3 | Page 26

$\tan \sqrt{x}$

Ex. 30.20 | Q 5.4 | Page 26

$\tan \sqrt{x}$

#### Chapter 30: Derivatives Exercise 30.30 solutions [Pages 33 - 34]

Ex. 30.30 | Q 1 | Page 33

x4 − 2 sin x + 3 cos

Ex. 30.30 | Q 2 | Page 33

3x + x3 + 33

Ex. 30.30 | Q 3 | Page 33

$\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}$

Ex. 30.30 | Q 4 | Page 33

ex log a + ea long x + ea log a

Ex. 30.30 | Q 5 | Page 33

(2x2 + 1) (3x + 2)

Ex. 30.30 | Q 6 | Page 33

log3 x + 3 loge x + 2 tan x

Ex. 30.30 | Q 7 | Page 34

$\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)$

Ex. 30.30 | Q 8 | Page 34

$\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3$

Ex. 30.30 | Q 9 | Page 34

$\frac{2 x^2 + 3x + 4}{x}$

Ex. 30.30 | Q 10 | Page 34

$\frac{( x^3 + 1)(x - 2)}{x^2}$

Ex. 30.30 | Q 11 | Page 34

$\frac{a \cos x + b \sin x + c}{\sin x}$

Ex. 30.30 | Q 12 | Page 34

2 sec x + 3 cot x − 4 tan x

Ex. 30.30 | Q 13 | Page 34

a0 xn + a1 xn−1 + a2 xn2 + ... + an1 x + an

Ex. 30.30 | Q 14 | Page 34

$\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}$

Ex. 30.30 | Q 15 | Page 34

$\frac{(x + 5)(2 x^2 - 1)}{x}$

Ex. 30.30 | Q 16 | Page 34

$\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}$

Ex. 30.30 | Q 17 | Page 34

cos (x + a)

Ex. 30.30 | Q 19 | Page 34

$\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .$

Ex. 30.30 | Q 20 | Page 34

$\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}$

Ex. 30.30 | Q 21 | Page 34

Find the slope of the tangent to the curve (x) = 2x6 + x4 − 1 at x = 1.

Ex. 30.30 | Q 22 | Page 34

$If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)$

Ex. 30.30 | Q 23 | Page 34

Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.

Ex. 30.30 | Q 24 | Page 34

$\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .$

Ex. 30.30 | Q 25 | Page 34

If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.

Ex. 30.30 | Q 26 | Page 34

For the function $f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .$

#### Chapter 30: Derivatives Exercise 30.40 solutions [Page 39]

Ex. 30.40 | Q 1 | Page 39

x3 sin

Ex. 30.40 | Q 2 | Page 39

x3 e

Ex. 30.40 | Q 3 | Page 39

x2 ex log

Ex. 30.40 | Q 4 | Page 39

xn tan

Ex. 30.40 | Q 5 | Page 39

xn loga

Ex. 30.40 | Q 6 | Page 39

(x3 + x2 + 1) sin

Ex. 30.40 | Q 7 | Page 39

sin x cos x

Ex. 30.40 | Q 8 | Page 39

$\frac{2^x \cot x}{\sqrt{x}}$

Ex. 30.40 | Q 9 | Page 39

x2 sin x log

Ex. 30.40 | Q 10 | Page 39

x5 ex + x6 log

Ex. 30.40 | Q 11 | Page 39

(x sin x + cos x) (x cos x − sin x

Ex. 30.40 | Q 12 | Page 39

(x sin x + cos x ) (ex + x2 log x

Ex. 30.40 | Q 13 | Page 39

(1 − 2 tan x) (5 + 4 sin x)

Ex. 30.40 | Q 14 | Page 39

(1 +x2) cos x

Ex. 30.40 | Q 15 | Page 39

sin2

Ex. 30.40 | Q 16 | Page 39

logx2 x

Ex. 30.40 | Q 17 | Page 39

$e^x \log \sqrt{x} \tan x$

Ex. 30.40 | Q 18 | Page 39

x3 ex cos

Ex. 30.40 | Q 19 | Page 39

$\frac{x^2 \cos\frac{\pi}{4}}{\sin x}$

Ex. 30.40 | Q 20 | Page 39

x4 (5 sin x − 3 cos x)

Ex. 30.40 | Q 21 | Page 39

(2x2 − 3) sin

Ex. 30.40 | Q 22 | Page 39

x5 (3 − 6x−9

Ex. 30.40 | Q 23 | Page 39

x4 (3 − 4x−5)

Ex. 30.40 | Q 24 | Page 39

x−3 (5 + 3x

Ex. 30.40 | Q 25 | Page 39

Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.

Ex. 30.40 | Q 26.1 | Page 39

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3x2 + 2)2

Ex. 30.40 | Q 26.2 | Page 39

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(x + 2) (x + 3)

Ex. 30.40 | Q 26.3 | Page 39

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)

Ex. 30.40 | Q 27 | Page 39

(ax + b) (a + d)2

Ex. 30.40 | Q 28 | Page 39

(ax + b)n (cx d)

#### Chapter 30: Derivatives Exercise 30.50 solutions [Page 44]

Ex. 30.50 | Q 1 | Page 44

$\frac{x^2 + 1}{x + 1}$

Ex. 30.50 | Q 2 | Page 44

$\frac{2x - 1}{x^2 + 1}$

Ex. 30.50 | Q 3 | Page 44

$\frac{x + e^x}{1 + \log x}$

Ex. 30.50 | Q 4 | Page 44

$\frac{e^x - \tan x}{\cot x - x^n}$

Ex. 30.50 | Q 5 | Page 44

$\frac{a x^2 + bx + c}{p x^2 + qx + r}$

Ex. 30.50 | Q 6 | Page 44

$\frac{x}{1 + \tan x}$

Ex. 30.50 | Q 7 | Page 44

$\frac{1}{a x^2 + bx + c}$

Ex. 30.50 | Q 8 | Page 44

$\frac{e^x}{1 + x^2}$

Ex. 30.50 | Q 9 | Page 44

$\frac{e^x + \sin x}{1 + \log x}$

Ex. 30.50 | Q 10 | Page 44

$\frac{x \tan x}{\sec x + \tan x}$

Ex. 30.50 | Q 11 | Page 44

$\frac{x \sin x}{1 + \cos x}$

Ex. 30.50 | Q 12 | Page 44

$\frac{2^x \cot x}{\sqrt{x}}$

Ex. 30.50 | Q 13 | Page 44

$\frac{\sin x - x \cos x}{x \sin x + \cos x}$

Ex. 30.50 | Q 14 | Page 44

$\frac{x^2 - x + 1}{x^2 + x + 1}$

Ex. 30.50 | Q 15 | Page 44

$\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}$

Ex. 30.50 | Q 16 | Page 44

$\frac{a + \sin x}{1 + a \sin x}$

Ex. 30.50 | Q 17 | Page 44

$\frac{{10}^x}{\sin x}$

Ex. 30.50 | Q 18 | Page 44

$\frac{1 + 3^x}{1 - 3^x}$

Ex. 30.50 | Q 19 | Page 44

$\frac{3^x}{x + \tan x}$

Ex. 30.50 | Q 20 | Page 44

$\frac{1 + \log x}{1 - \log x}$

Ex. 30.50 | Q 21 | Page 44

$\frac{4x + 5 \sin x}{3x + 7 \cos x}$

Ex. 30.50 | Q 22 | Page 44

$\frac{x}{1 + \tan x}$

Ex. 30.50 | Q 23 | Page 44

$\frac{a + b \sin x}{c + d \cos x}$

Ex. 30.50 | Q 24 | Page 44

$\frac{p x^2 + qx + r}{ax + b}$

Ex. 30.50 | Q 25 | Page 44

$\frac{\sec x - 1}{\sec x + 1}$

Ex. 30.50 | Q 26 | Page 44

$\frac{x^5 - \cos x}{\sin x}$

Ex. 30.50 | Q 27 | Page 44

$\frac{x + \cos x}{\tan x}$

Ex. 30.50 | Q 28 | Page 44

$\frac{x}{\sin^n x}$

Ex. 30.50 | Q 29 | Page 44

$\frac{ax + b}{p x^2 + qx + r}$

Ex. 30.50 | Q 30 | Page 44

$\frac{1}{a x^2 + bx + c}$

#### Chapter 30: Derivatives solutions [Pages 46 - 47]

Q 1 | Page 46

Write the value of $\lim_{x \to c} \frac{f(x) - f(c)}{x - c}$

Q 2 | Page 46

Write the value of $\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}$

Q 3 | Page 47

If x < 2, then write the value of $\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}$

Q 4 | Page 47

If $\frac{\pi}{2}$ then find $\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)$

Q 5 | Page 47

Write the value of $\frac{d}{dx}\left( x \left| x \right| \right)$

Q 6 | Page 47

Write the value of $\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}$

Q 7 | Page 47

If f (x) = |x| + |x−1|, write the value of $\frac{d}{dx}\left( f (x) \right)$

Q 8 | Page 47

Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.

Q 9 | Page 47

If f (x) = $\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)$

Q 10 | Page 47

Write the value of $\frac{d}{dx}\left( \log \left| x \right| \right)$

Q 11 | Page 47

If f (1) = 1, f' (1) = 2, then write the value of $\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}$

Q 12 | Page 47

Write the derivative of f (x) = 3 |2 + x| at x = −3.

Q 13 | Page 47

If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of $\frac{dy}{dx}$

Q 14 | Page 47

If f (x) =  $\log_{x_2}$write the value of f' (x).

#### Chapter 30: Derivatives solutions [Pages 47 - 48]

Q 1 | Page 47

Mark the correct alternative in of the following:

Let f(x) = x − [x], x ∈ R, then $f'\left( \frac{1}{2} \right)$

•  $\frac{3}{2}$

• 1

•  −1

Q 2 | Page 47

Mark the correct alternative in of the following:

If $f\left( x \right) = \frac{x - 4}{2\sqrt{x}}$

•  $\frac{5}{4}$

• $\frac{4}{5}$

•  1

•  0

Q 3 | Page 47

Mark the correct alternative in of the following:

If$y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .$then $\frac{dy}{dx} =$

•  y + 1

• y − 1

• y

•  y2

Q 4 | Page 48

Mark the correct alternative in  of the following:

If$f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}$then $f'\left( 1 \right)$

•  150

• −50

• −150

• 50

Q 5 | Page 48

Mark the correct alternative in  of the following:

If $y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}$ then $\frac{dy}{dx} =$

• $- \frac{4x}{\left( x^2 - 1 \right)^2}$

• $- \frac{4x}{x^2 - 1}$

• $\frac{1 - x^2}{4x}$

• $\frac{4x}{x^2 - 1}$

Q 6 | Page 48

Mark the correct alternative in of the following:

If $y = \sqrt{x} + \frac{1}{\sqrt{x}}$ then $\frac{dy}{dx}$ at x = 1 is

•  1

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

• 0

Q 7 | Page 48

Mark the correct alternative in of the following:
If $f\left( x \right) = x^{100} + x^{99} + . . . + x + 1$  then $f'\left( 1 \right)$ is equal to

• 5050

•  5049

• 5051

• 50051

Q 8 | Page 48

Mark the correct alternative in  of the following:
If$f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}$ then $f'\left( 1 \right)$ is equal to

• $\frac{1}{100}$

• 100

• 50

Q 9 | Page 48

Mark the correct alternative in each of the following:
If$y = \frac{\sin x + \cos x}{\sin x - \cos x}$ then $\frac{dy}{dx}$at x = 0 is

• −2

•  0

• $\frac{1}{2}$

• does not exist

Q 10 | Page 48

Mark the correct alternative in  of the following:
If $y = \frac{\sin\left( x + 9 \right)}{\cos x}$ then $\frac{dy}{dx}$ at x = 0 is

•  cos 9

• sin 9

•  0

• 1

Q 11 | Page 48

Mark the correct alternative in each of the following:

If$f\left( x \right) = \frac{x^n - a^n}{x - a}$ then $f'\left( a \right)$

•  1

•  0

• $\frac{1}{2}$

• does not exist

Q 12 | Page 48

Mark the correct alternative in of the following:

If f(x) = x sinx, then $f'\left( \frac{\pi}{2} \right) =$

• 1

• −1

• $\frac{1}{2}$

## Chapter 30: Derivatives

Ex. 30.10Ex. 30.20Ex. 30.30Ex. 30.40Ex. 30.50Others

## RD Sharma solutions for Class 11 Mathematics chapter 30 - Derivatives

RD Sharma solutions for Class 11 Maths chapter 30 (Derivatives) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics chapter 30 Derivatives are Limits of Exponential Functions, Derivative of Slope of Tangent of the Curve, Theorem for Any Positive Integer n, Graphical Interpretation of Derivative, Derive Derivation of x^n, Algebra of Derivative of Functions, Derivative of Polynomials and Trigonometric Functions, Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically, Limits of Logarithmic Functions, Intuitive Idea of Derivatives, Introduction of Limits, Introduction to Calculus, Algebra of Limits, Limits of Polynomials and Rational Functions, Introduction of Derivatives, Limits of Trigonometric Functions.

Using RD Sharma Class 11 solutions Derivatives exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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