KC Sinha Mathematics Solution Class 12 Chapter 11 अवकलन (Differentiation) Exercise 11.1 (Q21-Q24)

Exercise 11.1

Question 21

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

Differentiating with respect to x

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

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

$=\frac{\cos \sqrt{\sin \sqrt{x}} \cdot \cos \sqrt{x}}{4 \sqrt{x} \sqrt{\sin \sqrt{x}}}$

Question 22

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

Differentiating with respect to x

$\frac{d y}{d}=\frac{d(\sin \sqrt{\cos \sqrt{a x}})}{d(\sqrt{\cos \sqrt{a x})}} \times \frac{d(\sqrt{\cos \sqrt{a{x}}})}{d(\cos \sqrt{a{x}})}\times \frac{d(\cos \sqrt{a{x}}}{d(\sqrt{a{x}})} \times \frac{d (\sqrt{a{x})}}{dx}\times \frac{d(ax)}{dx}$

$=\cos \sqrt{\cos \sqrt{a x}}\times \frac{1}{2 \sqrt{\cos \sqrt{4 x}}} \times(-\sin \sqrt{a x}) \times \frac{1}{2 \sqrt{ax}}$

$=-\frac{1}{4} \times\frac{a}{\sqrt{a} \sqrt{x}} \cdot \frac{\sin \sqrt{a x} \cos \sqrt{\cos \sqrt{a x}}}{\sqrt{\cos \sqrt{ax}}}$

$=\frac{-1}{4} \sqrt{\frac{a}{x}} \cdot \frac{\sin \sqrt{a} x}{\sqrt{\cos \sqrt{a x}}} \times \cos \sqrt{\cos \sqrt{ax}}$

Question 23

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

Differentiating with respect to x

$\frac{d y}{d x}=\frac{d(\sqrt{\sin (\sqrt x)})}{d(\sin (\sin \sqrt{x}))} \cdot \frac{d(\sin (\sin \sqrt{x}))}{d(\sin \sqrt{x})} \times \frac{d(\sin \sqrt{x})}{d(\sqrt{2})}\times\frac{d(\sqrt{x})}{d x}$

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

$=\frac{\cos \sqrt{x} \cdot \cos (\sin\sqrt{x})}{4 \sqrt{x} \sqrt{\sin (\sin \sqrt{x})}}$

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

$\frac{d y}{d t}=-\sin (\tan \sqrt{x+1}) \cdot \sec ^{2} \sqrt{x+1} \times \frac{1}{2 \sqrt{x+1}}\times 1$

$=\frac{-\sec ^{2} \sqrt{x+1}. \sin (\tan \sqrt{x+1})}{2 \sqrt{x+1}}$

Question 24

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

Differentiating with respect to x