OP Malhotra Class 12 Maths Solutions Chapter 8 Differentiation Ex 8(d)

Students can cross-reference their work with S Chand ISC Maths Class 12 Solutions Chapter 8 Differentiation Ex 8(d) to ensure accuracy.

S Chand Class 12 ICSE Maths Solutions Chapter 8 Differentiation Ex 8(d)

Differentiate w.r.t. x :

Question 1.
(i) log cos x
(ii) log sin x
(iii) cos (log x)
(iv) \(\frac{1}{\log \cos x}\)
(v) x log x – x
Solution:
(i) Let y = log cos x
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{1}{\cos x} \frac{d}{d x} \cos x=-\frac{\sin x}{\cos x}\) = – tanx

(ii) Let y = log sin x
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{1}{\sin x} \frac{d}{d x} \sin x=\frac{\cos x}{\sin x}\) = cotx

(iii) Let y = cos (log x)
DifF both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{d}{d x} \cos (\log x)=-\sin (\log x) \frac{d}{d x} \log x\)
= – \(\frac{\sin (\log x)}{x}\)

(iv) \(\frac{1}{\log \cos x}\);
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{d}{d x}\left(\frac{1}{\log \cos x}\right)\)
= \(\frac{-1}{(\log \cos x)^2} \frac{d}{d x} \log \cos x\)
= \(\frac{-1}{(\log \cos x)^2} \frac{1}{\cos x}(-\sin x)\)
= \(\frac{\tan x}{(\log \cos x)^2}\)

(v) Let y = x log x – x
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=x \cdot \frac{d}{d x} \log x+\log x \frac{d}{d x} x-\frac{d}{d x} x\)
= x.\(\frac { 1 }{ x }\) + log x. 1-1 = 1 + log x – 1 = log x

Question 2.
(i) log \(\left(\sqrt[5]{x^3}\right)\)
(ii) log (3 – 7x)
(iii) log x³
(iv) log \(\sqrt{x}\)
(v) \(\frac{\sin x}{\log x}\)
Solution:
(i) log \(\left(\sqrt[5]{x^3}\right)\) = logx^3/5
= \(\frac { 3 }{ 5 }\)log x [∵ log a^b = b log a]
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{3}{5 x}\)

(ii) Let y = log (3 – 7x);
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{d}{d x} \log (3-7 x)\)
= \(\frac{1}{3-7 x} \frac{d}{d x}(3-7 x)=\frac{-7}{3-7 x}\)

(iii) Let y = log x³ = 3 log x
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{3}{x}\)

(iv) Let y = log \(\sqrt{x}\) = log x^1/2 = \(\frac { 1 }{ 2 }\) log x
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{1}{2 x}\)

(v) Let y = \(\frac{\sin x}{\log x}\)

Question 3.
log (cosec x – cot x)
Solution:

Question 4.
sin(log cos x)
Solution:
Let y = sin (log cos x)
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\cos (\log \cos x) \frac{d}{d x} \log \cos x\)
\(\cos (\log \cos x) \frac{1}{\cos x} \frac{d}{d x} \cos x\)
= \(\frac{\sin x}{\cos x}\)
= – tan x cos(log cos x)

Question 5.
log (log x)
Solution:
Let y = log(log x)
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{1}{\log x} \frac{d}{d x} \log x=\frac{1}{x \log x}\)

Question 6.
\(\frac{\log x}{1+\log x}\)
Solution:

Question 7.
sin (log x) – log sin x
Solution:
Let y = sin (log x) – log sin x
Diff both sides w.r.t. x ; we have
∴ \(\frac{d y}{d x}=\frac{d}{d x} \sin (\log x)-\frac{d}{d x}\) log(sin x)
= cos (log x).\(\frac { 1 }{ x }\) – \(\frac{1}{\sin x}\) cos x
= \(\frac{\cos (\log x)}{x}\) – cot x

Question 8.
\(\log \left(\frac{1-x^2}{1+x^2}\right)\)
Solution:

Question 9.
\(\log \sqrt{\frac{x-1}{x+1}}\)
Solution:

Question 10.
sin [sin (log 3x)]
Solution:
Let y = sin [sin (log 3x)]
Diff both sides w.r.t. x ; we have
∴ \(\frac{d y}{d x}=\cos (\sin (\log 3 x)) \frac{d}{d x} \sin (\log 3 x)\)
= \(\cos (\sin (\log 3 x)) \cos (\log 3 x) \frac{d}{d x} \log 3 x\)
= \(\cos (\sin (\log 3 x)) \cos (\log 3 x) \times \frac{1}{3 x} \times 3\)
= \(\frac{\cos [\sin (\log 3 x)] \cos (\log 3 x)}{x}\)

Question 11.
log cos \(\sqrt{x}\)
Solution:

Question 12.
cos (log x)²
Solution:
Let y = cos (log x)²
Diff both sides w.r.t. x ; we have
∴ \(\frac{d y}{d x}=-\sin (\log x)^2 \frac{d}{d x}\)(log x)²
= \(-\sin (\log x)^2 2 \log x \frac{d}{d x}(\log x)\)
= \(\frac{-2 \sin (\log x)^2 \log x}{x}\)

Question 13.
log \((\sqrt{\tan x})\)
Solution:

Question 14.
log \(\left(x+\sqrt{1+x^2}\right)\)
Solution:

Question 15.
\(\log \left(x-\sqrt{x^2-a^2}\right)\)
Solution:

Question 16.
sec x. tan x + log tan \(\left(\frac{\pi}{4}+\frac{x}{2}\right)\)
Solution:
sec x. tan x + log tan \(\left(\frac{\pi}{4}+\frac{x}{2}\right)\) ; Diff both sides w.r.t. x ; we have

Question 17.
log\(\sqrt{\frac{1+\sin x}{1-\sin x}}\)
Solution:

Question 18.
log\(\sqrt[3]{\frac{1-x}{1+x}}\)
Solution:

Question 19.
(ln ln x)²
Solution:

Question 20.
log\(\left(\sec \frac{x}{2}+\tan \frac{x}{2}\right)\)
Solution:

Question 21.
log [sin (log x)]
Solution:
Let y = log [sin (log x)]
Diff both sides w.r.t. x ; we have
\(\frac{d y}{d x}=\frac{1}{\sin (\log x)} \frac{d}{d x} \sin (\log x)\) = \(\frac{\cos (\log x)}{\sin (\log x)} \frac{d}{d x}(\log x)=\frac{\cot (\log x)}{x}\)

Question 22.
log\(\left[\log \left(\sin \sqrt{x^2+1}\right]\right.\)
Solution:

Question 23.
log\(\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}\)
Solution:

Question 24.
If y = \(\sqrt{x^2+1}-\log \left(\frac{1}{x}+\sqrt{1+\frac{1}{x^2}}\right)\), find \(\frac { dy }{ dx }\).
Solution: