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S Chand Class 12 ICSE Maths Solutions Chapter 8 Differentiation Ex 8(g)
Differentiate the following w.r.t. x :
Question 1.
(i) cos-1(cos x)
(ii) tan-1 (cot x)
Solution:
(i) Let y = cos-1(cos x) = x
[∵ cos-1(cos θ) = θ ∀ θ ∈ [ 0, π]]
Diff. both sides w.r.t. x ; we have
\(\frac { dy }{ dx }\) = 1
(ii) Let y = tan-1(cot x)
= \(\tan ^{-1}\left[\tan \left(\frac{\pi}{2}-x\right)\right]=\frac{\pi}{2}\) – x
Diff. both sides w.r.t. x ; we have
\(\frac { dy }{ dx }\) = 0 – 1 = – 1
Question 2.
(i) \(\tan ^{-1} \frac{\sin x}{1+\cos x}\)
(ii) \(\tan ^{-1} \frac{1-\cos x}{\sin x}\)
Solution:
Question 3.
\(\tan ^{-1} \frac{\cos x-\sin x}{\cos x+\sin x}\)
Solution:
Question 4.
\(\cot ^{-1}\left(\frac{\cos x}{1+\sin x}\right)\)
Solution:
Question 5.
\(\sin ^{-1} \sqrt{\frac{1-\cos 2 x}{2}}\)
Solution:
Let y = \(\sin ^{-1} \sqrt{\frac{1-\cos 2 x}{2}}\)
= \(\sin ^{-1} \sqrt{\frac{2 \sin ^2 x}{2}}=\sin ^{-1}(\sin x)\) = x
∴ \(\frac { dy }{ dx }\) = 1
Question 6.
\(\sin ^{-1}\left(\sqrt{1-x^2}\right)\)
Solution:
Question 7.
\(\sin ^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)\)
Solution:
Question 8.
\(\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\)
Solution:
Question 9.
\(\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)\)
Solution:
Question 10.
\(\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)\)
Solution:
Question 11.
cos-1 (2x² – 1)
Solution:
Let y = cos-1 (2x² – 1)
put x = cos θ ⇒ θ = cos-1x
∴ y = cos-1(2 cos²θ – 1)
∴ y = cos-1 (cos 2θ)
⇒ y = 2θ = 2 cos-1x
Diff. both sides w.r.t. x, we have
∴ \(\frac{d y}{d x}=\frac{-2}{\sqrt{1-x^2}}\)
Question 12.
sin-1 (3x – 4x³)
Solution:
Let y = sin-1 (3x² – 4x³)
put x = sin θ ⇒ θ = sin-1x
∴ y = sin-1(3sin θ – 4sin³ θ)
= sin-1(sin 3θ) = 3θ = 3
Diff. both sides w.r.t. x, we have
∴ \(\frac{d y}{d x}=\frac{3}{\sqrt{1-x^2}}\)
Question 13.
cot-1\(\left(\frac{1-x^2}{2 x}\right)\)
Solution:
Question 14.
cosec-1\(\left(\frac{1+x^2}{2 x}\right)\)
Solution:
Question 15.
\(\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\)
Solution:
Question 16.
\(\sec ^{-1}\left(\frac{x^2+1}{x^2-1}\right)\)
Solution:
Question 17.
(i) \(\tan ^{-1} \sqrt{\frac{1+x}{1-x}}\)
(ii) \(\cot ^{-1} \sqrt{\frac{1+x}{1-x}}\)
Solution:
Question 18.
(i) \(\cos ^{-1} \sqrt{\frac{1+x}{2}}\)
(ii) \(\cos ^{-1} \sqrt{\frac{1-x^2}{2}}\)
(iii) \(\sin ^{-1} \sqrt{\frac{1-x^2}{2}}\)
(iv) \(\cos ^{-1} \sqrt{\frac{1+x^2}{2}}\)
Solution:
Question 19.
\(\tan ^{-1} \frac{x}{\sqrt{\left(a^2-x^2\right)}}\)
Solution:
Question 20.
\(\tan ^{-1} \frac{x}{\sqrt{\left(a^2-x^2\right)}}\)
Solution:
Question 21.
\(\tan ^{-1} \frac{x}{1+\sqrt{1-x^2}}\)
Solution:
Question 22.
\(\tan ^{-1}\left(\frac{\sqrt{a}-\sqrt{x}}{1+\sqrt{a x}}\right)\)
Solution:
Question 23.
\(\tan ^{-1} \sqrt{\frac{2-x}{2+x}}\)
Solution:
Question 24.
\(\sin ^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)\)
Solution:
Question 25.
(i) \(\tan ^{-1}\left(\sqrt{1+x^2}+x\right)\)
(ii) \(\cot ^{-1}\left(\sqrt{1+x^2}-x\right)\)
Solution:
Question 26.
(i) \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}+1}{x}\right)\)
(ii) \(\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2-1}}{a x}\right)\)
Solution:
Question 27.
sin²\(\left[\cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right]\).
Solution:
