Question

If $y=e^{\sqrt{\cot x}},$ then $\frac{dy}{dx}=$

• A. $-\frac{e^{\sqrt{\cot x}}\times {\text{cosec}}^{2}x}{2\sqrt{\cot x}}$
• B. $\frac{e^{\sqrt{\cot x}}\times \text{cosec}x}{2\sqrt{\cot x}}$
• C. $-\frac{e^{\cot x}\times {\text{cosec}}^{2}x}{2\sqrt{\cot x}}$
• D. $-\frac{e^{\sqrt{\cot x}}\times {\text{cosec}}^{2}x}{\sqrt{\cot x}}$

Answer 1

The correct option is A $-\frac{e^{\sqrt{\cot x}}\times {\text{cosec}}^{2}x}{2\sqrt{\cot x}}$

We have,
$y=e^{\sqrt{\cot x}}$
$\Rightarrow y=e^{(\cot x{)}^{\frac{1}{2}}}$
Differentiate it with respect to $x$,
$\frac{dy}{dx}=\frac{d}{dx}\left(e^{(\cot x{)}^{\frac{1}{2}}}\right)$
$=e^{(\cot x{)}^{\frac{1}{2}}}\times \frac{d}{dx}\left(\cot x{)}^{\frac{1}{2}}\right[\text{Using chain rule}]$
$=e^{\sqrt{\cot x}}\times \frac{1}{2}\left(\cot x{)}^{\frac{1}{2}-1}\frac{d}{dx}\right(\cot x)$
$=-\frac{e^{\sqrt{\cot x}}\times {\text{cosec}}^{2}x}{2\sqrt{\cot x}}$