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

Class 11: Maths Chapter 30 solutions. Complete Class 11 Maths Chapter 30 Notes.

RD Sharma Solutions for Class 11 Maths Chapter 30–Derivatives

RD Sharma 11th Maths Chapter 30, Class 11 Maths Chapter 30 solutions

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,

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

Solution:

Given:

f(x) = x2 – 2

By using the derivative formula,

= 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,

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

Solution:

Given:

f(x) = x

By using the derivative formula,

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

Solution:

Given:

f(x) = cos x

By using the derivative formula,

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

Solution:

Given:

f(x) = tan x

By using the derivative formula,

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,

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

We know that 1 – cos x = 2 sin2(x/2)

(ii) x at x = 1

Given:

f (x) = x

By using the derivative formula,

(iii) 2 cos x at x = π/2

Given:

f (x) = 2 cos x

By using the derivative formula,

(iv) sin 2xat x = π/2

Solution:

Given:

f (x) = sin 2x

By using the derivative formula,

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

Now, multiply numerator and denominator by 2, we get

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,

∴ Derivative of f(x) = 2/x is -2x-2

(ii) 1/√x

Given:

f (x) = 1/√x

By using the formula,

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

(iii) 1/x3

Given:

f (x) = 1/x3

By using the formula,

∴ Derivative of f(x) = 1/x3 is -3x-4

(iv) [x2 + 1]/ x

Given:

f (x) = [x2 + 1]/ x

By using the formula,

= 1 – 1/x2

∴ Derivative of f(x) = 1 – 1/x2

(v) [x2 – 1] / x

Given:

f (x) = [x2 – 1]/ x

By using the formula,

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,

(ii) e3x

Given:

f (x) = e3x

By using the formula,

(iii) eax+b

Given:

f (x) = eax+b

By using the formula,

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,

(ii) sin x/x

Given:

f (x) = sin x/x

By using the formula,

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,

(ii) tan (2x + 1)

Given:

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

By using the formula,

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,

(ii) cos √x

Given:

f (x) = cos √x

f (x + h) = cos √(x+h)

By using the formula,

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

2. 3x + x3 + 33

Solution:

Given:

f (x) = 3x + x3 + 33

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

Solution:

Given:

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

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

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

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

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

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

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

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

EXERCISE 30.5 PAGE NO: 30.44

Differentiate the following functions with respect to x:

Solution:

Let us consider

y =

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

Solution:

Let us consider

y =

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

Solution:

Let us consider

y =

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

Solution:

Let us consider

y =

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

Solution:

Let us consider

y =

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: Download PDF

RD Sharma Solutions for Class 11 Maths Chapter 30–Derivatives

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About RD Sharma

RD Sharma isn’t the kind of author you’d bump into at lit fests. But his bestselling books have helped many CBSE students lose their dread of maths. Sunday Times profiles the tutor turned internet star
He dreams of algorithms that would give most people nightmares. And, spends every waking hour thinking of ways to explain concepts like ‘series solution of linear differential equations’. Meet Dr Ravi Dutt Sharma — mathematics teacher and author of 25 reference books — whose name evokes as much awe as the subject he teaches. And though students have used his thick tomes for the last 31 years to ace the dreaded maths exam, it’s only recently that a spoof video turned the tutor into a YouTube star.

R D Sharma had a good laugh but said he shared little with his on-screen persona except for the love for maths. “I like to spend all my time thinking and writing about maths problems. I find it relaxing,” he says. When he is not writing books explaining mathematical concepts for classes 6 to 12 and engineering students, Sharma is busy dispensing his duty as vice-principal and head of department of science and humanities at Delhi government’s Guru Nanak Dev Institute of Technology.

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