RBSE Maths Class 12 Chapter 7: Differentiation Important Questions and Solutions

RBSE Class 12 Maths Chapter 7 – Differentiation Important questions and solutions are provided here. All these important questions and solutions of the chapter given here are prepared by the subject experts at BYJU’S. The RBSE Class 12 important questions and solutions available at BYJU’S will help the students in scoring maximum marks in the Class 12 exams.

Chapter 7 of RBSE Class 12 has six exercises, all of them deal with differentiating the given functions of various types. In this chapter, students will learn differentiating various functions such as algebraic expressions, trigonometric functions, logarithm functions, exponential function and so on. All these exercises will help students to work on various word problems.

RBSE Maths Chapter 7: Exercise 7.1 Textbook Important Questions and Solutions

Question 1: Differentiate the following function with respect to x.

tan(2x + 3)

Solution:

Let the given function be y = tan (2x + 3)

Differentiating with respect to x,

dy/dx = d/dx (tan(2x + 3))

= sec²(2x + 3) d/dx(2x + 3)

= sec²(2x + 3).(2 × 1 + 0)

= 2 sec²(2x + 3)

Question 2: Differentiate the following function with respect to x.

sin{cos(x²)}

Solution:

Let the given function be y = sin {cos (x²)}

Differentiating with respect to x,

dy/dx = d/dx sin [cos(x²)]

= cos(cos x²). (- sin x²) d/dx (x²)

= -cos(cos x²) (sin x²). 2x

= -2x (sin x²) cos(cos x²)

RBSE Maths Chapter 7: Exercise 7.2 Textbook Important Questions and Solutions

Question 3: Differentiate the following function with respect to x.

(a) sin^-1{2x √(1 – x²)}, -1/√2 < x < 1/√2

(b) sin^-1(3x – 4x³) x ∈ (-1/2, 1/2)

Solution:

(a) sin^-1{2x √(1 – x²)}, -1/√2 < x < 1/√2

Let y = sin^-1{2x √(1 – x²)}

Substitute x = sin θ

y = sin^-1{2 sin θ √(1 – sin²θ)}

= sin^-1(2 sin θ cos θ)

= sin^-1(sin 2θ)

= 2θ

y = 2 sin^-1(x)

Differentiate with respect to x,

dy/dx = d/dx (2 sin^-1x)

2 d/dx(sin^-1x)

= 2[1/√(1 – x²)]

dy/dx = 2/√(1 – x²)

(b) sin^-1(3x – 4x³) x ∈ (-1/2, 1/2)

Let y = sin^-1(3x – 4x³)

Substituting x = sin θ,

θ = sin^-1x

Thus, y = sin^-1(3 sin θ – 4 sin³θ)

= sin^-1(sin 3θ)

= 3θ

= 3 sin^-1x

Differentiate with respect to x,

dy/dx = d/dx (3 sin^-1x)

= 3 d/dx (sin^-1x)

= 3[1/√(1 – x²)]

dy/dx = 3/√(1 – x²)

Question 4: Differentiate the following function with respect to x.

(a) sec-1[1/(2x² – 1)]; x ∈ (0, 1/√2)

(b) cos-1[(1 – x²)/ (1 + x²)]; x ∈ (0, ∞)

Solution:

(a) sec-1[1/(2x² – 1)]; x ∈ (0, 1/√2)

Let y = sec^-1[1/(2x² – 1)]

Substituting x = cos θ,

y = sec^-1[1/(2 cos²θ – 1)]

= sec^-1(1/ cos 2θ)

= sec^-1(sec 2θ)

y = 2θ

y = 2 cos^-1x

Differentiating with respect to x,

dy/dx = d/dx (2 cos^-1x)

= 2 d/dx(cos^-1x)

= 2[-1/√(1 – x²)]

dy/dx = -2/√(1 – x²)

(b) cos-1[(1 – x²)/ (1 + x²)]; x ∈ (0, ∞)

Let y = cos^-1[(1 – x²)/ (1 + x²)]

Substituting x = tan θ

y = cos^-1[(1 – tan²θ)/ (1 + tan²θ)

= cos^-1(cos 2θ)

y = 2θ

y = 2 tan^-1x

Differentiating with respect to x,

dy/dx = d/dx(2 tan^-1x)

= 2[1/(1 + x²)]

dy/dx = 2/(1 + x²)

RBSE Maths Chapter 7: Exercise 7.3 Textbook Important Questions and Solutions

Question 5: Find dy/dx.

(a) 2x + 3y = sin y

(b) x² + xy + y² = 200

Solution:

(a) 2x + 3y = sin y

Differentiate with respect to x on both sides,

d/dx(2x) + d/dx(3y) = d/dx(sin y)

2(1) + 3(dy/dx) = cos y (dy/dx)

2 = (cosy – 3) dy/dx

Therefore, dy/dx = 2/(cos y – 3)

(b) x² + xy + y² = 200

Differentiate with respect to x on both sides,

d/dx (x²) + d/dx (xy) + d/dx (y²) = d/dx(200)

2x + x(dy/dx) + 2y (dy/dx) = 0

(x + 2y)dy/dx = -2x – y

Therefore, dy/dx = -(2x + y)/(x + 2y)

Question 6: Find dy/dx.

(a) √x + √y = √a

(b) tan(x + y) + tan(x – y) = 4

Solution:

(a) √x + √y = √a

Differentiating with respect to x,

d/dx(√x) + d/dx(√y) = d/dx(√a)

dy/dx = -(√y)/(√x)

Therefore, dy/dx = -√(y/x)

(ii) tan(x + y) + tan(x – y) = 4

Differentiating with respect to x,

d/dx [tan(x + y)] + d/dx [tan(x – y)] = d/dx (4)

sec²(x + y) d/dx(x + y) + sec²(x – y)d/dx(x – y) = 0

sec²(x + y) [1 + (dy/dx)] + sec²(x – y)[1 – (dy/dx)] = 0

sec²(x + y) + sec²(x + y)dy/dx + sec²(x – y) – sec²(x – y)dy/dx = 0

-[sec²(x + y) – sec²(x + y)] dy/dx = -[sec²(x + y) + sec²(x – y)]

Therefore, dy/dx = [sec²(x + y) + sec²(x – y)]/ [sec²(x – y) – sec²(x + y)]

RBSE Maths Chapter 7: Exercise 7.4 Textbook Important Questions and Solutions

Question 7: Find dy/dx, when x = log t, y = e^t + cos t.

Solution:

Given,

x = log t, y = e^t + cos t

Consider, x = log t

Differentiating with respect to t on both sides,

dx/dt = d/dt (log t)

dx/dt = 1/t

Now, y = e^t + cos t

Differentiating with respect to t on both sides,

dy/dt = d/dt(e^t) + d/dt(cos t)

dy/dt = e^t – sin t

dy/dt = (dy/dt)/ (dx/dt)

= (e^t – sin t)/ (1/t)

= t(e^t – sin t)

Therefore, dy/dt = t(e^t – sin t)

Question 8: Find dy/dx, when x = a cos θ, y = b sin θ.

Solution:

Given,

x = a cos θ, y = b sin θ

Consider, x = a cos θ

Differentiating with respect to θ on both sides,

dx/dθ = d/dθ(a cos θ)

dx/dθ = -a sin θ

Now, y = b sin θ

Differentiating with respect to θ on both sides,

dy/dθ = d/dθ(b sin θ)

dy/dθ = b cos θ

dy/dx = (dy/dθ)/ (dx/dθ)

= (b cos θ)/ (-a sin θ)

= -(b/a) cot θ

Therefore, dy/dx = -(b/a) cot θ

RBSE Maths Chapter 7: Exercise 7.5 Textbook Important Questions and Solutions

Question 9: Find d²y/dx², when y = x² + 3x + 2.

Solution:

Given,

y = x² + 3x + 2

Differentiating with respect to x on both sides,

dy/dx = d/dx(x² + 3x + 2)

dy/dx = 2x + 3

Again, differentiating with respect to x on both sides,

d/dx(dy/dx) = d/dx(2x + 3)

d²y/dx² = 2(1) + 0

Therefore, d²y/dx² = 2

Question 10: If y = a cos nx + b sin nx, then prove that d²y/dx² + n²y = 0.

Solution:

Given,

y = a cos nx + b sin nx

Differentiating with respect to x on both sides,

dy/dx = d/dx(a cos nx + b sin nx)

= -na sin nx + nb cos nx

Again, differentiating with respect to x on both sides,

d/dx(dy/dx) = d/dx(-na sin nx + nb cos nx)

d²y/dx² = -n²a cos nx – n²b sin nx

= -n²(a cos nx + b sin nx)

= -n²y (from the given)

Therefore, d²y/dx² + n²y = 0

Hence proved.

RBSE Maths Chapter 7: Exercise 7.6 Textbook Important Questions and Solutions

Question 11: Verify Rolle’s theorem for the functions given below.

f(x) = cos 2x; x ∈ [0, π]

Solution:

Given function

f(x) = cos 2x, x ∈ [0, π]

⇒ f(x) = cos 2x is defined in the interval [0, π].

We know that the cosine is continuous in this domain.

Thus, it is continuous in [0, π].

Therefore, f'(x) = -2 sin 2x exists wherein x ∈ (0, π)

⇒ f(x) is differentiable in (0, π).

Now,

f(0) = cos 0 = 1

f(π) = cos 2π = 1

∴ f(0) = f(π) = 1

Thus, all the conditions of Rolle’s theorem are satisfied hence, at least one point c such that c ∈ (0, π) and f'(c) = 0.

f'(c) = 0

-2 sin 2c = 0

sin 2c = 0

2c = π

c = π/2 is an element of (0,π)

∴ c = π/2 ∈ (0,π) such that f'(c) = 0

Hence, for c = π/2, Rolle’s theorem is satisfied.

Question 12: Verify Lagrange’s mean value theorem for the function given below.

f(x) = x² – 3x + 2; x ∈ [-2, 3]

Solution:

Given function is:

f(x) = x2 – 3x + 2, x ∈ [- 2, 3]

Thus, f(x) is continuous in the interval [-2, 3].

Hence, f'(x) is finite and exists in (-2, 3).

That means, f(x) is differentiable in (- 2, 3).

Therefore, f(x) satisfies both the conditions of Langrange’s mean value theorem.

f'(c) = [f(b) – f(a)]/(b – a)

2c – 3 = [f(3) – f(-2)]/ [3 – (-2)]

2c – 3 = (2 – 12)/5

2c – 3 = -10/2

2c – 3 = -2

2c = -2 + 3

2c = 1

c = 1/2

Thus, c = 1/2 ∈ (-2, 3)

Hence, the Langrange’s mean value theorem is satisfied.

RBSE Maths Chapter 7: Additional Important Questions and Solutions

Question 1: Differentiate log (x/a^x) with respect to x.

Solution:

Let the given function be:

y = log(x/ax)

y = log x – log ax

y = log x – x log a

Differentiating with respect to x on both sides,

dy/dx = d/dx(log x) – d/dx(x log a)

dy/dx = (1/x) – log a (1)

Therefore, dy/dx = (1/x) – log a

Question 2: If sin y = x sin(a + y), then prove that dy/dx = sin²(a + y)/sin a.

Solution:

Given function is:

sin y = x sin(a + y)

x = sin y/sin(a + y)

Differentiating with respect to y on both sides,

dx/dy = [sin(a + y). cos y – sin y. cos(a + y)]/ sin²(a + y)

dx/dy = [sin(a + y – y)]/ sin²(a + y)

dx/dy = sin a/sin²(a + y)

Therefore, dy/dx = sin²(a + y)/sin a