KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q37-Q40)
Sol :
Let y=$\sqrt{\frac{1+\sin x}{1-\sin x}}$
$y=\sqrt{\frac{1+\sin x}{1-\sin x} \times \frac{1+\sin x}{1+\sin x}}$
$y=\sqrt{\frac{(1+\sin x)^{2}}{1^2-\sin ^{2} x}}$
$y=\sqrt{\frac{(1+\sin x)^{2}}{\cos ^{2} x}}$
$y=\frac{1+\sin x}{\cos x}$
$y=\frac{1}{\cos x}+\frac{\sin x}{\cos x}$
y=sec x+tan x
Differentiating with respect to x
$\frac{d y}{d x}=\sec x \tan x+\sec ^{2} x$
=sec x(tan x+sec x)
Question 38
Sol :
Let y=$\left(\frac{2 \tan x}{\tan x+\cos x}\right)^{2}$
Differentiating with respect to x
$\frac{d y}{dx}=\frac{d\left(\frac{2 \tan x}{\tan x+\cos x}\right)^{2}}{d\left(\frac{2 \tan x}{\tan x+\cos x}\right)} \times\frac{d \left(\frac{2 \tan x}{\tan x+\cos x}\right)}{d x}$
$=2\left(\frac{2 \tan x}{\tan x+\cos x}\right) \cdot \frac{2\left[\sec ^{2} x \cdot(\tan x+\cos x)-\tan \left(\sec ^{2} x-\sin x)\right]\right.}{(\tan x+\cos x)^{2}}$
$=\frac{8 \tan x\left[\sec ^{2} x\tan x+\sec^{2} x \cos x-\sec^{2} x \tan x+ \tan x \sin x\right.]}{(\tan x+\cos x)^3}$
$=\frac{8 \tan x \cdot\left(\sec ^{2} x \cos x+\tan x \sin x\right)}{(\tan x+\cos x)^{3}}$
$=\frac{8 \tan x(\sec x+\tan x \sin x)}{(\tan x+\cos x)^{3}}$
Question 39
Sol :
Let y=$\sqrt{x} \sin x+\sin \sqrt{x}$
Differentiating with respect to x
$\frac{d y}{dx}=d\left(\frac{\sqrt{x} \sin x}{d x}\right)+\frac{d(\sin \sqrt{x})}{d x}$
$=\frac{d(\sqrt{x})}{d} \cdot \sin x+\sqrt{x} \cdot \frac{d(\sin x)}{dx}+\frac{d(\sin \sqrt{x})}{d(\sqrt{x})} \times \frac{d(\sqrt{x})}{dx}$
$=\frac{1}{2 \sqrt{x}} \sin x+\sqrt{x} \cos x+\frac{1}{2 \sqrt{x}} \cos \sqrt{x}$
$=\frac{1}{2 \sqrt{2}}\left[\sin x+2 x \cos x+\cos \sqrt{x}\right]$
Question 40
Sol :
Let y=$\cos \left(a x^{2}+b x+c\right)+\sin ^{3} \sqrt{a x^{2}+b x+c}$
Differentiating with respect to x
$\frac{d y}{d x}=\frac{d\left[\cos\left(a x^{2}+b x+c\right)\right]}{d\left(a x^{2}+b x+c\right)} \cdot \frac{d\left(a x^{2}+b x+c\right)}{d x}+\frac{d(\sin^ 3 \sqrt{ax^ 2+b x+c})}{d(\sin \sqrt{a x^{2}+b x+c})} \times \frac{d(\sin \sqrt{a x^{2}+b x+c})}{d(\sqrt{ax^{2}+b x+c})}\times \frac{d(\sqrt{a x^{2}+b x+c})}{d\left(a x^{2}+b x+c\right)} \times \frac{d\left(a x^{2}+b x+c\right)}{d x}$
$+3 \sin ^{2} \sqrt{a x^{2}+b x+c} \times \cos \sqrt{a x^{2}+b x+c}\times \frac{1}{2 \sqrt{ax^2+bc+c}}\times(2ax+b)$
$=-(2 a x+b) \sin \left(a x^{2}+b x+c\right)+\frac{3}{2} \frac{(2 a x+b) \cdot \cos \sqrt{a x^{2}+b x+c} \cdot \sin ^{2} \sqrt{a x^{2}+b x+c}}{\sqrt{a x^{2}+b x+c}}$
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