RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 Textbook Exercise Questions and Answers.

Rajasthan Board RBSE Solutions for Class 12 Maths in Hindi Medium & English Medium are part of RBSE Solutions for Class 12. Students can also read RBSE Class 12 Maths Important Questions for exam preparation. Students can also go through RBSE Class 12 Maths Notes to understand and remember the concepts easily.

RBSE Class 12 Maths Solutions Chapter 5 Continuity and Differentiability Ex 5.2

Question 1.
sin (x2 + 5)
Answer:
Let y = sin (x2 + 5)
∴ Differentiating both sides w.r.t. x, we gel
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\) sin (x2 + 5)
= cos (x2 + 5). \(\frac{d}{d x}\)(x2 + 5)
= cos(x2 + 5).(2x)
\(\frac{d}{d x}\)[sin(x2 + 5)] = 2x cos (x2 + 5)

RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Question 2.
cos (sin x)
Answer:
Let y = cos (sin x)
Differentiating both sides w.r.t. ‘x’, we get
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\) [cos(sin x)]
\(\frac{d y}{d x}\) = - sin(sin x). \(\frac{d}{d x}\)(sin x)
\(\frac{d}{d x}\)[cos (sin x) = - sin (sin x) cos x

Question 3.
sin (ax + b)
Answer:
Let y = sin (ax + b)
y = sin (ax + b)
Differentiating both sides w.r.t. ‘x’, we get
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)[sin (ax + b)]
\(\frac{d y}{d x}\) = cos(ax + b) \(\frac{d}{d x}\) (ax + b)
\(\frac{d y}{d x}\) = cos (ax + b) × (a.1 × 0)
= cos (ax + b).(a)
\(\frac{d y}{d x}\) = a cos (ax + b)

RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Question 4.
sec (tan √x)
Answer:
Let y = sec [tan (√x)]
Differentiating both sides w.r.t. ‘x’, we get
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 1

Question 5.
\(\frac{\sin (a x+b)}{\cos (c x+d)}\)
Answer:
Let y = \(\frac{\sin (a x+b)}{\cos (c x+d)}\)
Differentiating both sides w.r.t. ‘x’, we get
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 2
= a cos (ax + b) sec (cx + d) + c sin (ax + d) sec (cx + d) tan (cx + d)

RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Question 6.
cos x3.sin2 (x5)
Answer:
Let y = cos x3.sin2 (x5)
Differentiating both sides w.r.t. ‘x’, we get
\(\frac{d y}{d x}\) = cos x3 \(\frac{d}{d x}\) {sin2 (x5)} + sin2 (x5) \(\frac{d}{d x}\) (cos x3)
= cos x3.2 sin x5.cos x5. \(\frac{d}{d x}\) (x5) + sin2 x5.(- sin x3).\(\frac{d}{d x}\)(x3)
\(\frac{d y}{d x}\) = cos x3.2 sin (x5).cos (x5)
= 10x4 cos x3 cos x5 - 3x2 sin x3.sin2 x5

Question 7.
2\(\sqrt{\cot (x)^{2}}\)
Answer:
Let y = 2\(\sqrt{\cot (x)^{2}}\)
Differentiating both sides w.r.t. ‘x’, we get
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 3

Question 8.
cos √x
Answer:
Let y = cos √x
Differentiating both sides w.r.t. 'x', we get
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 4

RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Question 9.
Prove that the function f given by f(x) = |x - 1|; x ∈ R, is not differentiable at x = 1.
Answer:
f(x) = x + 1
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 5
∴ L.H.D. = - 1 ≠ 1 = R.H.D.
Thus, function is not Differentiable at x = 1.
Hence Proved.

Question 10.
Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3, is not differentiable at x = 1 and x = 2.
Answer:
(i) For x = 1
Left hand derivative
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 6
[Since 1 is greater integer before (1 + h)]
∵ derivative of left side \(\lim _{h \rightarrow 0}\frac{1}{h}\) is not defined.
Thus, function is not differentiable at x = 1.
Hence Proved.

RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

(ii) For x = 2
Left hand derivative
RBSE Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 7
[Since maximum integer in 2 before (2 + h)]
∵ Derivative of LHS \(\lim _{h \rightarrow 0}\frac{1}{h}\) is not defined.
Thus, fraction is not differentiable at x = 2.
Hence Proved.

Bhagya
Last Updated on Nov. 2, 2023, 9:25 a.m.
Published Nov. 1, 2023