Exercise 11.1
Question 21
Sol :
Let y=$\sin \sqrt{\sin \sqrt{x}}$
$\frac{d y}{d x}=\frac{d[\sin \sqrt{\sin \sqrt{x}}]}{d(\sqrt{\sin \sqrt{x}})} \times \frac{d(\sqrt{\sin \sqrt{x}})}{d(\sin \sqrt{x})} \times \frac{d(\sin \sqrt{x})}{d(\sqrt{x})}\times \frac{d(\sqrt x)}{dx}$
$=\cos \sqrt{\sin \sqrt{x}} \cdot \frac{1}{2 \sqrt{\sin \sqrt{x}}} \times \cos \sqrt{x} \cdot \frac{1}{2 \sqrt{x}}$
$=\frac{\cos \sqrt{\sin \sqrt{x}} \cdot \cos \sqrt{x}}{4 \sqrt{x} \sqrt{\sin \sqrt{x}}}$
Question 22
Sol :
Let y=$\sin \sqrt{\cos \sqrt{a x}}$
$\frac{d y}{d}=\frac{d(\sin \sqrt{\cos \sqrt{a x}})}{d(\sqrt{\cos \sqrt{a x})}} \times \frac{d(\sqrt{\cos \sqrt{a{x}}})}{d(\cos \sqrt{a{x}})}\times \frac{d(\cos \sqrt{a{x}}}{d(\sqrt{a{x}})} \times \frac{d (\sqrt{a{x})}}{dx}\times \frac{d(ax)}{dx}$
$=\cos \sqrt{\cos \sqrt{a x}}\times \frac{1}{2 \sqrt{\cos \sqrt{4 x}}} \times(-\sin \sqrt{a x}) \times \frac{1}{2 \sqrt{ax}}$
$=-\frac{1}{4} \times\frac{a}{\sqrt{a} \sqrt{x}} \cdot \frac{\sin \sqrt{a x} \cos \sqrt{\cos \sqrt{a x}}}{\sqrt{\cos \sqrt{ax}}}$
$=\frac{-1}{4} \sqrt{\frac{a}{x}} \cdot \frac{\sin \sqrt{a} x}{\sqrt{\cos \sqrt{a x}}} \times \cos \sqrt{\cos \sqrt{ax}}$
Question 23
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
Sol:
Let y=$\cos (\tan \sqrt{x+1})$
Differentiating with respect to x
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