Question

The derivative of $\cot \sqrt{x}$ is

• A. $\frac{-{\sin }^2\sqrt{x}}{2\sqrt{x}}$
• B. $\frac{{\sin }^2\sqrt{x}}{2\sqrt{x}}$
• C. $\frac{-{\text{cosec}}^2\sqrt{x}}{2\sqrt{x}}$
• D. $\frac{-{\sec }^2\sqrt{x}}{2\sqrt{x}}$

Answer 1

Answer: (C)

Let $f(x)=\cot \sqrt{x}$.
Then, $f(x+h)=\cot \sqrt{x+h}$
$\therefore \frac{d}{dx}f(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$
$\Rightarrow \frac{d}{dx}f(x)=\underset{h\to 0}{\lim }\frac{\cot \sqrt{x+h}-\cot \sqrt{x}}{h}$
$\Rightarrow \frac{d}{dx}(f(x))=\underset{h\to 0}{\lim }\frac{-\sin (\sqrt{x+h}-\sqrt{x})}{h\sin \sqrt{x+h}\sin \sqrt{x}}$
$\Rightarrow \frac{d}{dx}f(x)=\underset{h\to 0}{\lim }\frac{-\sin (\sqrt{x+h}-\sqrt{x})}{[(x+h)-x]\sin \sqrt{x+h}\sin \sqrt{x}]}$
$\Rightarrow \frac{d}{dx}f(x)=\underset{h\to 0}{\lim }\frac{-\sin (\sqrt{x+h}-\sqrt{x})}{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}x\sin \sqrt{x+h}\sin \sqrt{x}$
$\Rightarrow \frac{d}{dx}f(x)=\underset{h\to 0}{\lim }\frac{-\sin (\sqrt{x+h}-\sqrt{x})}{\sqrt{x+h}-\sqrt{x}}\times \underset{h\to 0}{\lim }\frac{1}{(\sqrt{x+h}+\sqrt{x})\sin \sqrt{x+h}\sin \sqrt{x}}$
$\Rightarrow \frac{d}{dx}f(x)=\frac{-1}{2\sqrt{x}\sin \sqrt{x}\sin \sqrt{x}}$
$\Rightarrow \frac{d}{dx}f(x)=\frac{-{\mathrm{cosec}}^2\sqrt{x}}{2\sqrt{x}}$