KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q17-Q20)

Exercise 11.1

Question 17

$\sqrt{\tan (\tan x)}$
Sol :
Let y=$\sqrt{\tan (\tan x)}$

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d(\sqrt{\tan (\tan x)})}{d(\tan (\tan x))}=\frac{d(\tan (\tan x))}{d(\tan x)} \frac{d(\tan x)}{dx}$

$=\frac{1}{2 \sqrt{\tan (\tan x)}} \times \sec^{2}(\tan x) \cdot \sec ^{2} x$

$=\frac{\sec ^{2} x \cdot \sec ^{2}(\tan x)}{2 \sqrt{\tan (\tan x)}}$

Question 18

$\sqrt{1+\sin x}$
Sol :
Let y=$\sqrt{1+\sin x}$

Differentiating with respect to x

$\frac{d{y}}{dx}=\frac{d(\sqrt{1+\sin x})}{d(1+\sin x)} \times \frac{d\left(1+\sin x\right)}{d x}$

$=\frac{1}{2 \sqrt{1+\sin x}} \times \cos x$

$=\frac{\cos x}{2 \sqrt{1+\sin x}}$

Question 19

$\sqrt{\tan \left(1+x^{2}\right)}$
Sol :
Let y=$\sqrt{\tan \left(1+x^{2}\right)}$

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d(\sqrt{\tan \left(1+x^{2}\right)})}{d\left(\tan \left(1+x^{2}\right)\right)} \times \left.\frac{\left.d(\tan \left(1+x^{2}\right)\right)}{d\left(1+x^{2}\right)} \times \frac{d\left(1+x^{2}\right)}{dx}\right.$

$=\frac{1}{2 \sqrt{\tan \left(1+x^{2}\right)}} \times \sec ^{2}\left(1+x^{2}\right) \cdot 2 x$

$=\frac{x \sec ^{2}\left(1+x^{2}\right)}{\sqrt{\tan \left(1+x^{2}\right)}}$

Question 20

$\cot \sqrt{\cos \sqrt{x}}$
Sol :
Let y=$\cot \sqrt{\cos \sqrt{x}}$

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d[\cot\sqrt{\cos \sqrt{x}})}{d[\sqrt{\cos \sqrt{x}}]} \times \frac{d \sqrt{\cos \sqrt{x}}}{d(\cos \sqrt{x})} \times \frac{d( \cos \sqrt{x})}{d(\sqrt{x})}\times\frac{d(\sqrt{x})}{dx}$

$=-\operatorname{cosec}^{2} \sqrt{\cos \sqrt{x}}\times \frac{1}{2 \sqrt{\cos \sqrt{x}}} \times (-\sin \sqrt{x}) \times \frac{1}{2 \sqrt{x}}$

$=\frac{\sin \sqrt{x} \operatorname{cosec}^{2} \sqrt{\cos \sqrt{x}}}{4 \sqrt{x} \sqrt{\cos \sqrt{x}}}$