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

### 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,

**2. Find the derivative of f(x) = x ^{2}Â â€“ 2 at x = 10**

**Solution:**

Given:

f(x) = x^{2}Â â€“ 2

By using the derivative formula,

= 0 + 20 = 20

Hence,

Derivative of f(x) = x^{2}Â â€“ 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 = Ï€/2(ii) x at x = 1**

**(iii) 2 cos x at x = Ï€/2**

**(iv) sin 2xat x = Ï€/2**

**Solution:**

**(i) sin x at x = Ï€/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 â€“ cos x = 2 sin^{2}(x/2)

**(ii) x at x = 1**

Given:

f (x) = x

By using the derivative formula,

**(iii) 2 cos x at x = Ï€/2**

Given:

f (x) = 2 cos x

By using the derivative formula,

**(iv) sin 2xat x = Ï€/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/âˆšx**

**(iii) 1/x ^{3}**

**(iv) [x ^{2} + 1]/ x**

**(v) [x ^{2} – 1] / x**

**Solution:**

**(i) **2/x

Given:

f (x) = 2/x

By using the formula,

âˆ´ Derivative of f(x) = 2/x is -2x^{-2}

**(ii) **1/âˆšx

Given:

f (x) = 1/âˆšx

By using the formula,

âˆ´ Derivative of f(x) = 1/âˆšx is -1/2 x^{-3/2}

**(iii) 1/x ^{3}**

Given:

f (x) = 1/x^{3}

By using the formula,

âˆ´ Derivative of f(x) = 1/x^{3} is -3x^{-4}

**(iv) **[x^{2} + 1]/ x

Given:

f (x) = [x^{2} + 1]/ x

By using the formula,

= 1 â€“ 1/x^{2}

âˆ´ Derivative of f(x) =Â 1 â€“ 1/x^{2}

**(v) **[x^{2} – 1] / x

Given:

f (x) = [x^{2} – 1]/ x

By using the formula,

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

**(i) e ^{-x}**

**(ii) e ^{3x}**

**(iii) e ^{ax+b}**

**Solution:**

**(i) **e^{-x}

Given:

f (x) = e^{-x}

By using the formula,

**(ii) **e^{3x}

Given:

f (x) = e^{3x}

By using the formula,

**(iii) **e^{ax+b}

Given:

f (x) = e^{ax+b}

By using the formula,

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

**(i) âˆš(sin 2x)**

**(ii) sin x/x**

**Solution:**

**(i) **âˆš(sin 2x)

Given:

f (x) = âˆš(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) tan ^{2} x **

**(ii) tan (2x + 1)**

**Solution:**

**(i) **tan^{2} x** **

Given:

f (x) = tan^{2} 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 âˆš2x**

**(ii) cos âˆšx**

**Solution:**

**(i) **sin âˆš2x

Given:

f (x) = sin âˆš2x

f (x + h) = sin âˆš2(x+h)

By using the formula,

**(ii) **cos âˆšx

Given:

f (x) = cos âˆšx

f (x + h) = cos âˆš(x+h)

By using the formula,

#### EXERCISE 30.3 PAGE NO: 30.33

**Differentiate the following with respect to x:**

**1. x ^{4}Â â€“ 2sin x + 3 cos x**

**Solution:**

Given:

f (x) = x^{4}Â â€“ 2sin x + 3 cos x

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

**2.** **3 ^{x}Â + x^{3}Â + 3^{3}**

**Solution:**

Given:

f (x) = 3^{x}Â + x^{3}Â + 3^{3}

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. e ^{x log a}Â + e^{a log x}Â + e^{a log a}**

**Solution:**

Given:

f (x) = e^{x log a}Â + e^{a log x}Â + e^{a log a}

We know that,

e^{log f(x)} =Â f(x)

So,

f(x) = a^{x}Â + x^{a}Â + a^{a}

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

**5. (2x ^{2}Â + 1) (3x + 2)**

**Solution:**

Given:

f (x) = (2x^{2}Â + 1) (3x + 2)

Â = 6x^{3}Â + 4x^{2}Â + 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. x ^{3}Â sin x**

**Solution:**

Let us consider y = x^{3}Â sin x

We need to find dy/dx

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

u = x^{3}Â and v = sin x

âˆ´Â y = uv

Now let us apply product rule of differentiation.

By using product rule, we get

**2. x ^{3}Â e^{x}**

**Solution:**

Let us consider y = x^{3}Â e^{x}

We need to find dy/dx

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

u = x^{3}Â and v = e^{x}

âˆ´Â y = uv

Now let us apply product rule of differentiation.

By using product rule, we get

**3. x ^{2}Â e^{x}Â log x**

**Solution:**

Let us consider y = x^{2}Â e^{x}Â log x

We need to find dy/dx

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

u = x^{2}Â and v = e^{x}

âˆ´Â y = uv

Now let us apply product rule of differentiation.

By using product rule, we get

**4. x ^{n}Â tan x**

**Solution:**

Let us consider y = x^{n}Â tan x

We need to find dy/dx

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

u = x^{n}Â and v = tan x

âˆ´Â y = uv

Now let us apply product rule of differentiation.

By using product rule, we get

**5. x ^{n}Â log_{a}Â x**

**Solution:**

Let us consider y = x^{n}Â log_{a}Â x

We need to find dy/dx

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

u = x^{n}Â and v = log_{a}Â x

âˆ´Â 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 = x^{2} + 1Â and v = x + 1

âˆ´Â 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 – 1Â and v = x^{2} + 1

âˆ´Â 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 + e^{x}Â and v = 1 + log x

âˆ´Â 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 = e^{x} â€“ tan xÂ and v = cot x – x^{n}

âˆ´Â 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 = ax^{2} + bx + cÂ and v = px^{2} + qx + r

âˆ´Â y = u/v

Now let us apply quotient rule of differentiation.

By using quotient rule, we get