Question

Differentiate

$e^{\sqrt{\cot x}}$

Answer 1

$$\begin{gathered} \mathrm{Let}y=e^{\sqrt{\cot x}} \\ \Rightarrow y=e^{{\left(\cot x\right)}^{\frac{1}{2}}} \\ \mathrm{Differentiate}\mathrm{it}\mathrm{with}\mathrm{respect}\mathrm{to}x\mathrm{we}\mathrm{get}, \\ \frac{\mathit{d}y}{\mathit{d}x}=\frac{d}{dx}\left(e^{{\left(\cot x\right)}^{\frac{1}{2}}}\right) \\ =e^{{\left(\cot x\right)}^{\frac{1}{2}}}\times \frac{d}{dx}{\left(\cot x\right)}^{\frac{1}{2}}\left[\mathrm{Using}\mathrm{chain}\mathrm{rule}\right] \\ =e^{\sqrt{\cot x}}\times \frac{1}{2}{\left(\cot x\right)}^{\frac{1}{2}-1}\frac{d}{dx}\left(\cot x\right) \\ =-\frac{e^{\sqrt{\cot x}}\times {\mathrm{cosec}}^{2}x}{2\sqrt{\cot x}} \\ \mathrm{So},\frac{d}{dx}\left(e^{\sqrt{\cot \mathrm{x}}}\right)=-\frac{e^{\sqrt{\cot x}}\times {\mathrm{cosec}}^{2}x}{2\sqrt{\cot x}} \end{gathered}$$