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Question
Differentiate \[e^\sqrt{\cot x}\] ?
Solution
\[\text{Let y} = e^{\sqrt{\cot x} }\]
\[\Rightarrow y = e^{\left( \cot x \right)^\frac{1}{2}} \]
\[\text{Differentiate it with respect to x we get}, \]
\[\frac{d y}{d x} = \frac{d}{dx}\left( e^{\left( \cot x \right)^\frac{1}{2} }\right)\]
\[ = e^{\left( \cot x \right)^\frac{1}{2}} \times \frac{d}{dx} \left( \cot x \right)^\frac{1}{2} .....\left[ \text{ Using chain rule} \right]\]
\[ = e^\sqrt{\cot x} \times \frac{1}{2} \left( \cot x \right)^{\frac{1}{2} - 1} \frac{d}{dx}\left( \cot x \right)\]
\[ = - \frac{e^\sqrt{\cot x} \times {cosec}^2 x}{2\sqrt{\cot x}}\]
\[So, \frac{d}{dx}\left( e^\sqrt{\cot x} \right) = - \frac{e^\sqrt{\cot x} \times {cosec}^2 x}{2\sqrt{\cot x}}\]
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