KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q25-Q28)

Exercise 11.1

Question 25

$\sin \sqrt{\cos \sqrt{\tan m x}}$
Sol :
Let y=$\sin \sqrt{\cos \sqrt{\tan m x}}$

Differentiating with respect to x

$\frac{d y}{d}=\frac{d(\sin \sqrt{\cos \sqrt{\tan m x}})}{d(\sqrt{\cos \sqrt{\tan m x}})} \times \frac{d(\sqrt{\cos \sqrt{\tan m x}})}{d(\cos \sqrt{\tan m x})}\times\frac{{d}(\cos \sqrt{m x})}{d(\sqrt{\tan m x})}\times \frac{d(\sqrt{\tan mx})}{d(mx)}\times \frac{d(\tan mx)}{d(mx)}\times\frac{d(mx)}{dx}$

$=\cos \sqrt{\cos \sqrt{\tan mx}} \cdot \frac{1}{2 \sqrt{\cos \sqrt{\tan m x}}} \times - \sin \sqrt{\tan m x} \times \frac{1}{2 \sqrt{\tan mx}\times \sec^2 mx \times m}$

$=\frac{-m \sec ^{2} m x \sin \sqrt{\tan m x} \cdot \cos \sqrt{\cos \sqrt{\tan mx}}}{4 \sqrt{\tan mx} \cdot \sqrt{\cos \sqrt{\tan m x}}}$

Question 26

$\frac{1}{\left(1+\tan ^{3} x\right)^{2}}$
Sol :
Let $y=\frac{1}{\left(1+\tan ^{3} x\right)^{2}}$

$y=\left(1+\tan ^{3} x\right)^{-2}$

Differentiating with respect to x

$\frac{d y}{dx}=\frac{d\left[\left(1+\tan^{3} x\right)^{-2}\right]}{d[1+\tan ^3 x]} \times \frac{d\left(1+\tan ^{3} x\right)}{d x}$

$=-2\left(1+\tan ^{3} x\right)^{-3} \cdot\left[\frac{d(1)}{dx}+\frac{d\left(\tan ^{3} x\right)}{d(\tan x)} \times \frac{d(\tan x)}{dx}\right]$

$=\frac{-2}{\left(1+\tan ^{3} x\right)^{3}} \cdot 3 \tan ^{2} x \cdot \sec ^{2} x$

$=\frac{-6 \tan ^{2} x \cdot \sec ^{2} x}{\left(1+\tan ^{3} x\right)^{3}}$

Question 27

$\cos \left(\frac{1-x^{2}}{1+x^{2}}\right)$
Sol :
Let y=$\cos \left(\frac{1-x^{2}}{1+x^{2}}\right)$

Differentiating with respect to x

$\frac{dy}{dx}=\frac{d\left[\cos \left(\frac{1-x^{2}}{1-x^{2}}\right)\right]}{d\left(\frac{1-x^{2}}{1+x^{2}}\right)} \times \frac{d\left(\frac{\left(-x^{2}\right)}{1+x^{2}}\right)}{dx}$

$=-\sin \left(\frac{1-x^{2}}{1+x^{2}}\right) \times \frac{\frac{d\left(1-x^{2}\right)}{d} \times\left(1+x^{2}\right)-\left(1-x^{2}\right) \times \frac{d(1+x^2)}{2}}{\left(1+x^{2}\right)^{2}}$

$=-\sin \left(\frac{1-x^{2}}{1+x^{2}}\right) \cdot \frac{-2 x\left(1+x^{2}\right)-\left(1-x^{2}\right){2 x}}{\left(1+x^{2}\right)^{2}}$

$=-\sin \left(\frac{1-x^{2}}{1+x^{2}}\right) \cdot \frac{-2 x-2 x^{3}-2 x+2 x^{3}}{\left(1+x^{2}\right)^{2}}$

$=\frac{4 x}{\left(1+x^{2}\right)^{2}} \cdot \sin \left(\frac{1-x^{2}}{1+x^{2}}\right)$

Question 28

$\cos \left(\frac{x}{1+\sqrt{x}}\right)$
Sol :
Let y=$\cos \left(\frac{x}{1+\sqrt{x}}\right)$

Differentiating with respect to x

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

$=-\sin \left(\frac{x}{1+\sqrt{x}}\right) \cdot \frac{\frac{d \cdot x}{d x}(1+\sqrt{x})-x \cdot \frac{d(1+\sqrt{x})}{dx}}{(1+\sqrt{x})^{2}}$

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

$=-\sin \left(\frac{x}{1+\sqrt{x}}\right) \cdot \frac{\frac{2+2 \sqrt{x}-\sqrt{x}}{2}}{(1+\sqrt{x})^{2}}$

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