Continuous practice using S Chand ISC Maths Class 12 Solutions Chapter 8 Differentiation Ex 8(f) can lead to a stronger grasp of mathematical concepts.
S Chand Class 12 ICSE Maths Solutions Chapter 8 Differentiation Ex 8(f)
Differentiate w.r.t. x the following function :
Question 1.
sin-1(3x)
Solution:
Let y = sin-13x,; diff. both sides w.r.t. x, we have
Question 2.
\(\cos ^{-1}\left(\frac{x}{2}\right)\)
Solution:
Question 3.
sin-1(2x – 3)
Solution:
Let y = sin-1(2x – 3) ; diff. both sides w.r.t. x, we have
\(\frac{d y}{d x}=\frac{1}{\sqrt{1-(2 x-3)^2}} \frac{d}{d x}(2 x-3)\)
= \(\frac{2}{\sqrt{-4 x^2+12 x-8}}=\frac{1}{\sqrt{-x^2+3 x-2}}\)
Question 4.
tan-13x²
Solution:
Let y = tan-13x²; diff. both sides w.r.t. x, we have
Question 5.
tan-13x²
Solution:
Let y = tan-13x²; diff. both sides w.r.t. x, we have
\(\frac{d y}{d x}=\frac{1}{\sqrt{1-(\sqrt{x})^2}} \times \frac{1}{2 \sqrt{x}}\)
\(\frac{1}{2 \sqrt{x-x^2}}\)
Question 6.
\(\cos ^{-1} \frac{x}{1+x}\)
Solution:
Question 7.
(sin-1 x)³
Solution:
Let y = (sin-1 x)³ ; diff. both sides w.r.t. x, we have
\(\frac{d y}{d x}=3\left(\sin ^{-1} x\right)^2 \frac{d}{d x} \sin ^{-1} x\)
= \(\frac{3\left(\sin ^{-1} x\right)^2}{\sqrt{1-x^2}}\)
Question 8.
\(\cot ^{-1} \frac{1+x}{1-x}\)
Solution:
Question 9.
cos-1(sin x)
Solution:
Let y = cos-1 (sin x)
Question 10.
sin(tan-1x)
Solution:
Let y = sin(tan-1x) ; Diff. both sides w.r.t. A, we have
\(\frac{d y}{d x}=\cos \left(\tan ^{-1} x\right) \cdot \frac{d}{d x} \tan ^{-1} x\)
= \(\frac{1}{1+x^2} \cos \left(\tan ^{-1} x\right)\)
Question 11.
x²cos-1\(\frac { 2 }{ x }\)
Solution:
Question 12.
\(\frac{1}{a b} \tan ^{-1}\left(\frac{b}{a} \tan x\right)\)
Solution:
Question 13.
\(\frac{x \cos ^{-1} x}{\sqrt{1-x^2}}\)
Solution:
Question 14.
\(x \sqrt{a^2-x^2}+a^2 \sin ^{-1} \frac{x}{a}\)
Solution:
Question 15.
If y = sin (2 sin-1 x), show that \(\frac{d y}{d x}=\frac{2 \sqrt{\left(1-y^2\right)}}{\sqrt{1-x^2}}\)
Solution:
Let y = sin (2 sin-1 x); Diff. both sides w.r.t. x, we have
