KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q29-Q32)

Exercise 11.1

Question 29

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

Differentiating with respect to x

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

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

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

$=\frac{2}{2 \sqrt{x}(1-\sqrt{x})^{2}}$

$=\frac{1}{\sqrt{x}(1-\sqrt{x})^{2}}$

Question 30

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

Differentiating with respect to x

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

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

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

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

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

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

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

Question 31

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

Differentiating with respect to x

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

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

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

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

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

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

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

Question 32

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

Differentiating with respect to x

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

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

$-\frac{(\cos x-\sin x) \cdot \cos \sqrt{\sin x+\cos x}}{2 \sqrt{\sin x+\cos x}}$