RD Sharma Solutions for Class 12 Maths Chapter 11–Differentiation

Class 12: Maths Chapter 11 solutions. Complete Class 12 Maths Chapter 11 Notes.

RD Sharma Solutions for Class 12 Maths Chapter 11–Differentiation

RD Sharma 12th Maths Chapter 11, Class 12 Maths Chapter 11 solutions

Exercise 11.1 Page No: 11.17

Differentiate the following functions from the first principles:

1. e^-x

Solution:

2. e^3x

Solution:

3. e^{ax + b}

Solution:

4. e^{cos x}

Solution:

We have to find the derivative of e^cos x with the first principle method,

So, let f (x) = e^cos x

By using the first principle formula, we get,

Solution:

Exercise 11.2 Page No: 11.37

Differentiate the following functions with respect to x:

1. Sin (3x + 5)

Solution:

Given Sin (3x + 5)

2. tan² x

Solution:

Given tan² x

3. tan (x^o + 45^o)

Solution:

Let y = tan (x° + 45°)

First, we will convert the angle from degrees to radians.

4. Sin (log x)

Solution:

Given sin (log x)

Solution:

6. e^{tan x}

Solution:

7. Sin² (2x + 1)

Solution:

Let y = sin² (2x + 1)

On differentiating y with respect to x, we get

8. log₇ (2x – 3)

Solution:

9. tan 5x^o

Solution:

Let y = tan (5x°)

First, we will convert the angle from degrees to radians. We have

Solution:

Solution:

12. log_x 3

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

18. (log sin x)²

Solution:

Let y = (log sin x)²

Solution:

Solution:

21. e^3x cos 2x

Solution:

22. Sin (log sin x)

Solution:

23. e^{tan 3x}

Solution:

Solution:

Solution:

Solution:

27. tan (e^{sin x})

Solution:

Solution:

Solution:

30. log (cosec x – cot x)

Solution:

Solution:

Solution:

33. tan^-1 (e^x)

Solution:

Solution:

35. sin (2 sin^-1 x)

Solution:

Let y = sin (2sin^–1x)

On differentiating y with respect to x, we get

Solution:

Solution:

Exercise 11.3 Page No: 11.62

Differentiate the following functions with respect to x:

Solution:

Solution:

Solution:

Let,

Solution:

Let,

Solution:

Solution:

7. Sin^-1 (2x² – 1), 0 < x < 1

Solution:

8. Sin^-1 (1 – 2x²), 0 < x < 1

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Let,

Solution:

Solution:

Solution:

Solution:

Exercise 11.4 Page No: 11.74

Find dy/dx in each of the following:

1. xy = c²

Solution:

2. y³ – 3xy² = x³ + 3x²y

Solution:

Given y³ – 3xy² = x³ + 3x²y,

Now we have to find dy/dx of given equation, so by differentiating the equation on both sides with respect to x, we get,

3. x^2/3 + y^2/3 = a^2/3

Solution:

Given x^2/3 + y^2/3 = a^2/3,

Now we have to find dy/dx of given equation, so by differentiating the equation on both sides with respect to x, we get,

4. 4x + 3y = log (4x – 3y)

Solution:

Given 4x + 3y = log (4x – 3y),

Now we have to find dy/dx of it, so by differentiating the equation on both sides with respect to x, we get,

Solution:

6. x⁵ + y⁵ = 5xy

Solution:

Given x⁵ + y⁵ = 5xy

Now we have to find dy/dx of given equation, so by differentiating the equation on both sides with respect to x, we get,

7. (x + y)² = 2axy

Solution:

Given (x + y)² = 2axy

Now we have to find dy/dx of given equation, so by differentiating the equation on both sides with respect to x, we get,

8. (x² + y²)² = xy

Solution:

Given (x + y)² = 2axy

Now we have to find dy/dx of given equation, so by differentiating the equation on both sides with respect to x, we get,

9. Tan^-1 (x² + y²)

Solution:

Given tan ^{– 1}(x² + y²) = a,

Now we have to find dy/dx of given function, so by differentiating the equation on both sides with respect to x, we get,

Solution:

11. Sin xy + cos (x + y) = 1

Solution:

Given Sin x y + cos (x + y) = 1

Now we have to find dy/dx of given function, so by differentiating the equation on both sides with respect to x, we get,

Solution:

Solution:

Solution:

Solution:

Exercise 11.5 Page No: 11.88

Differentiate the following functions with respect to x:

1. x^1/x

Solution:

2. x^{sin x}

Solution:

3. (1 + cos x)^x

Solution:

Solution:

5. (log x)^x

Solution:

6. (log x)^{cos x}

Solution:

Let y = (log x)^{cos x}

Taking log both the sides, we get

7. (Sin x)^{cos x}

Solution:

8. e^{x log x}

Solution:

9. (Sin x)^{log x}

Solution:

10. 10^{log sin x}

Solution:

11. (log x)^{log x}

Solution:

Solution:

13. Sin (x^x)

Solution:

14. (Sin^-1 x)^x

Solution:

Solution:

16. (tan x)^1/x

Solution:

Solution:

18. (i) (x^x) √x

Solution:

Solution:

Solution:

Solution:

Solution:

18. (vi) e^{sin x} + (tan x)^x

Solution:

18. (vii) (cos x)^x + (sin x)^1/x

Solution:

Solution:

19. y = e^x + 10^x + x^x

Solution:

20. y = xⁿ + n^x + x^x + nⁿ

Solution:

Exercise 11.6 Page No: 11.98

Solution:

Solution:

Solution:

Solution:

Exercise 11.7 Page No: 11.103

Find dy/dx, when

1. x = at² and y = 2 at

Solution:

2. x = a (θ + sin θ) and y = a (1 – cos θ)

Solution:

3. x = a cos θ and y = b sin θ

Solution:

Given x = a cos θ and y = b sin θ

4. x = a e^θ (sin θ – cos θ), y = a e^θ (sin θ + cos θ)

Solution:

5. x = b sin² θ and y = a cos² θ

Solution:

6. x = a (1 – cos θ) and y = a (θ + sin θ) at θ = π/2

Solution:

Solution:

Solution:

9. x = a (cos θ + θ sin θ) and y = a (sin θ – θ cos θ)

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Exercise 11.8 Page No: 11.112

1. Differentiate x² with respect to x³.

Solution:

2. Differentiate log (1 +x²) with respect to tan^-1 x.

Solution:

3. Differentiate (log x)^x with respect to log x.

Solution:

4. Differentiate sin^-1 √ (1-x²) with respect to cos^-1x, if

(i) x ∈ (0, 1)

(ii) x ∈ (-1, 0)

Solution:

(i) Given sin^-1 √ (1-x²)

(ii) Given sin^-1 √ (1-x²)

Solution:

(i) Let

(ii) Let

(iii) Let

Solution:

(i) x ∈ (0, 1/ √2)

(ii) x ∈ (1/√2, 1)

Solution:

(i) Let

(ii) Let

8. Differentiate (cos x)^{sin x} with respect to (sin x)^{cos x}.

Solution:

Solution:

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 11: Download PDF

RD Sharma Solutions for Class 12 Maths Chapter 11–Differentiation

Download PDF: RD Sharma Solutions for Class 12 Maths Chapter 11–Differentiation PDF

Chapterwise RD Sharma Solutions for Class 12 Maths :

• Chapter 1–Relation
• Chapter 2–Functions
• Chapter 3–Binary Operations
• Chapter 4–Inverse Trigonometric Functions
• Chapter 5–Algebra of Matrices
• Chapter 6–Determinants
• Chapter 7–Adjoint and Inverse of a Matrix
• Chapter 8–Solution of Simultaneous Linear Equations
• Chapter 9–Continuity
• Chapter 10–Differentiability
• Chapter 11–Differentiation
• Chapter 12–Higher Order Derivatives
• Chapter 13–Derivatives as a Rate Measurer
• Chapter 14–Differentials, Errors and Approximations
• Chapter 15–Mean Value Theorems
• Chapter 16–Tangents and Normals
• Chapter 17–Increasing and Decreasing Functions
• Chapter 18–Maxima and Minima
• Chapter 19–Indefinite Integrals

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.