# RD Sharma solutions for Class 11 Mathematics Textbook chapter 30 - Derivatives [Latest edition]

#### Chapters ## Chapter 30: Derivatives

Exercise 30.1Exercise 30.2Exercise 30.3Exercise 30.4Exercise 30.5Others
Exercise 30.1 [Page 3]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives Exercise 30.1 [Page 3]

Exercise 30.1 | Q 1 | Page 3

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

Exercise 30.1 | Q 2 | Page 3

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

Exercise 30.1 | Q 3 | Page 3

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

Exercise 30.1 | Q 4 | Page 3

Find the derivative of f (xx at x = 1

Exercise 30.1 | Q 5 | Page 3

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

Exercise 30.1 | Q 6 | Page 3

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

Exercise 30.1 | Q 7.1 | Page 3

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

sin x at x =$\frac{\pi}{2}$

Exercise 30.1 | Q 7.2 | Page 3

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

Exercise 30.1 | Q 7.3 | Page 3

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

2 cos x at x =$\frac{\pi}{2}$

Exercise 30.1 | Q 7.4 | Page 3

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

sin 2x at x =$\frac{\pi}{2}$

Exercise 30.2 [Pages 25 - 26]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives Exercise 30.2 [Pages 25 - 26]

Exercise 30.2 | Q 1.01 | Page 25

$\frac{2}{x}$

Exercise 30.2 | Q 1.02 | Page 25

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

Exercise 30.2 | Q 1.03 | Page 25

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

Exercise 30.2 | Q 1.04 | Page 25

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

Exercise 30.2 | Q 1.05 | Page 25

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

Exercise 30.2 | Q 1.06 | Page 25

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

Exercise 30.2 | Q 1.07 | Page 25

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

Exercise 30.2 | Q 1.08 | Page 25

k xn

Exercise 30.2 | Q 1.09 | Page 25

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

Exercise 30.2 | Q 1.1 | Page 25

x2 + x + 3

Exercise 30.2 | Q 1.11 | Page 25

(x + 2)3

Exercise 30.2 | Q 1.12 | Page 25

(x2 + 1) (x − 5)

Exercise 30.2 | Q 1.13 | Page 25

(x2 + 1) (x − 5)

Exercise 30.2 | Q 1.14 | Page 25

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

Exercise 30.2 | Q 1.15 | Page 25

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

Exercise 30.2 | Q 2.01 | Page 25

Differentiate each of the following from first principle:

ex

Exercise 30.2 | Q 2.02 | Page 25

Differentiate  of the following from first principle:

e3x

Exercise 30.2 | Q 2.03 | Page 25

Differentiate  of the following from first principle:

eax + b

Exercise 30.2 | Q 2.04 | Page 25

x ex

Exercise 30.2 | Q 2.05 | Page 25

Differentiate  of the following from first principle:

− x

Exercise 30.2 | Q 2.06 | Page 25

Differentiate of the following from first principle:

(−x)−1

Exercise 30.2 | Q 2.07 | Page 25

Differentiate  of the following from first principle:

sin (x + 1)

Exercise 30.2 | Q 2.08 | Page 25

Differentiate  of the following from first principle:

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

Exercise 30.2 | Q 2.09 | Page 25

Differentiate  of the following from first principle:

x sin x

Exercise 30.2 | Q 2.1 | Page 25

Differentiate of the following from first principle:

x cos x

Exercise 30.2 | Q 2.11 | Page 25

Differentiate  of the following from first principle:

sin (2x − 3)

Exercise 30.2 | Q 3.01 | Page 26

Differentiate each of the following from first principle:

$\sqrt{\sin 2x}$

Exercise 30.2 | Q 3.02 | Page 26

Differentiate each of the following from first principle:

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

Exercise 30.2 | Q 3.03 | Page 26

Differentiate each of the following from first principle:

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

Exercise 30.2 | Q 3.04 | Page 26

Differentiate each of the following from first principle:

x2 sin x

Exercise 30.2 | Q 3.05 | Page 26

Differentiate each of the following from first principle:

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

Exercise 30.2 | Q 3.06 | Page 26

Differentiate each of the following from first principle:

sin x + cos x

Exercise 30.2 | Q 3.07 | Page 26

Differentiate each of the following from first principle:

x2 e

Exercise 30.2 | Q 3.08 | Page 26

Differentiate each of the following from first principle:

$e^{x^2 + 1}$

Exercise 30.2 | Q 3.09 | Page 26

Differentiate each  of the following from first principle:

$e^\sqrt{2x}$

Exercise 30.2 | Q 3.1 | Page 26

Differentiate each of the following from first principle:

$e^\sqrt{ax + b}$

Exercise 30.2 | Q 3.11 | Page 26

Differentiate each of the following from first principle:

$a^\sqrt{x}$

Exercise 30.2 | Q 3.12 | Page 26

Differentiate each of the following from first principle:

$3^{x^2}$

Exercise 30.2 | Q 4.1 | Page 26

tan2

Exercise 30.2 | Q 4.2 | Page 26

tan (2x + 1)

Exercise 30.2 | Q 4.3 | Page 26

tan 2

Exercise 30.2 | Q 4.4 | Page 26

$\sqrt{\tan x}$

Exercise 30.2 | Q 5.1 | Page 26

$\sin \sqrt{2x}$

Exercise 30.2 | Q 5.2 | Page 26

$\cos \sqrt{x}$

Exercise 30.2 | Q 5.3 | Page 26

$\tan \sqrt{x}$

Exercise 30.2 | Q 5.4 | Page 26

$\tan \sqrt{x}$

Exercise 30.3 [Pages 33 - 34]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives Exercise 30.3 [Pages 33 - 34]

Exercise 30.3 | Q 1 | Page 33

x4 − 2 sin x + 3 cos

Exercise 30.3 | Q 2 | Page 33

3x + x3 + 33

Exercise 30.3 | Q 3 | Page 33

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

Exercise 30.3 | Q 4 | Page 33

ex log a + ea long x + ea log a

Exercise 30.3 | Q 5 | Page 33

(2x2 + 1) (3x + 2)

Exercise 30.3 | Q 6 | Page 33

log3 x + 3 loge x + 2 tan x

Exercise 30.3 | Q 7 | Page 34

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

Exercise 30.3 | Q 8 | Page 34

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

Exercise 30.3 | Q 9 | Page 34

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

Exercise 30.3 | Q 10 | Page 34

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

Exercise 30.3 | Q 11 | Page 34

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

Exercise 30.3 | Q 12 | Page 34

2 sec x + 3 cot x − 4 tan x

Exercise 30.3 | Q 13 | Page 34

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

Exercise 30.3 | Q 14 | Page 34

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

Exercise 30.3 | Q 15 | Page 34

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

Exercise 30.3 | Q 16 | Page 34

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

Exercise 30.3 | Q 17 | Page 34

cos (x + a)

Exercise 30.3 | Q 19 | Page 34

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

Exercise 30.3 | Q 20 | Page 34

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

Exercise 30.3 | Q 21 | Page 34

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

Exercise 30.3 | Q 22 | Page 34

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

Exercise 30.3 | Q 23 | Page 34

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

Exercise 30.3 | Q 24 | Page 34

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

Exercise 30.3 | 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.

Exercise 30.3 | Q 26 | Page 34

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

Exercise 30.4 [Page 39]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives Exercise 30.4 [Page 39]

Exercise 30.4 | Q 1 | Page 39

x3 sin

Exercise 30.4 | Q 2 | Page 39

x3 e

Exercise 30.4 | Q 3 | Page 39

x2 ex log

Exercise 30.4 | Q 4 | Page 39

xn tan

Exercise 30.4 | Q 5 | Page 39

xn loga

Exercise 30.4 | Q 6 | Page 39

(x3 + x2 + 1) sin

Exercise 30.4 | Q 7 | Page 39

sin x cos x

Exercise 30.4 | Q 8 | Page 39

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

Exercise 30.4 | Q 9 | Page 39

x2 sin x log

Exercise 30.4 | Q 10 | Page 39

x5 ex + x6 log

Exercise 30.4 | Q 11 | Page 39

(x sin x + cos x) (x cos x − sin x

Exercise 30.4 | Q 12 | Page 39

(x sin x + cos x ) (ex + x2 log x

Exercise 30.4 | Q 13 | Page 39

(1 − 2 tan x) (5 + 4 sin x)

Exercise 30.4 | Q 14 | Page 39

(1 +x2) cos x

Exercise 30.4 | Q 15 | Page 39

sin2

Exercise 30.4 | Q 16 | Page 39

logx2 x

Exercise 30.4 | Q 17 | Page 39

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

Exercise 30.4 | Q 18 | Page 39

x3 ex cos

Exercise 30.4 | Q 19 | Page 39

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

Exercise 30.4 | Q 20 | Page 39

x4 (5 sin x − 3 cos x)

Exercise 30.4 | Q 21 | Page 39

(2x2 − 3) sin

Exercise 30.4 | Q 22 | Page 39

x5 (3 − 6x−9

Exercise 30.4 | Q 23 | Page 39

x4 (3 − 4x−5)

Exercise 30.4 | Q 24 | Page 39

x−3 (5 + 3x

Exercise 30.4 | 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.

Exercise 30.4 | 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

Exercise 30.4 | 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)

Exercise 30.4 | 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)

Exercise 30.4 | Q 27 | Page 39

(ax + b) (a + d)2

Exercise 30.4 | Q 28 | Page 39

(ax + b)n (cx d)

Exercise 30.5 [Page 44]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives Exercise 30.5 [Page 44]

Exercise 30.5 | Q 1 | Page 44

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

Exercise 30.5 | Q 2 | Page 44

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

Exercise 30.5 | Q 3 | Page 44

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

Exercise 30.5 | Q 4 | Page 44

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

Exercise 30.5 | Q 5 | Page 44

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

Exercise 30.5 | Q 6 | Page 44

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

Exercise 30.5 | Q 7 | Page 44

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

Exercise 30.5 | Q 8 | Page 44

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

Exercise 30.5 | Q 9 | Page 44

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

Exercise 30.5 | Q 10 | Page 44

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

Exercise 30.5 | Q 11 | Page 44

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

Exercise 30.5 | Q 12 | Page 44

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

Exercise 30.5 | Q 13 | Page 44

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

Exercise 30.5 | Q 14 | Page 44

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

Exercise 30.5 | Q 15 | Page 44

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

Exercise 30.5 | Q 16 | Page 44

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

Exercise 30.5 | Q 17 | Page 44

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

Exercise 30.5 | Q 18 | Page 44

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

Exercise 30.5 | Q 19 | Page 44

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

Exercise 30.5 | Q 20 | Page 44

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

Exercise 30.5 | Q 21 | Page 44

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

Exercise 30.5 | Q 22 | Page 44

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

Exercise 30.5 | Q 23 | Page 44

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

Exercise 30.5 | Q 24 | Page 44

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

Exercise 30.5 | Q 25 | Page 44

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

Exercise 30.5 | Q 26 | Page 44

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

Exercise 30.5 | Q 27 | Page 44

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

Exercise 30.5 | Q 28 | Page 44

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

Exercise 30.5 | Q 29 | Page 44

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

Exercise 30.5 | Q 30 | Page 44

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

[Pages 46 - 47]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives [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).

[Pages 47 - 48]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 30 Derivatives [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

Exercise 30.1Exercise 30.2Exercise 30.3Exercise 30.4Exercise 30.5Others ## RD Sharma solutions for Class 11 Mathematics Textbook chapter 30 - Derivatives

RD Sharma solutions for Class 11 Mathematics Textbook 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 Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide 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 Textbook 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.

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