A) \[\frac{1}{4}{{e}^{x/2}}\cot ({{e}^{x/2}})\]
B) \[{{e}^{x/2}}\cot ({{e}^{x/2}})\]
C) \[\frac{1}{4}{{e}^{x}}\cot \,({{e}^{x}})\]
D) \[\frac{1}{2}{{e}^{x/2}}\cot \,({{e}^{x/2}})\]
Correct Answer: A
Solution :
\[\frac{d}{dx}[\log \sqrt{\sin \sqrt{{{e}^{x}}}}]=\frac{d}{dx}\left[ \frac{1}{2}\log (\sin \sqrt{{{e}^{x}}}) \right]\] \[=\frac{1}{2}\cot \sqrt{{{e}^{x}}}\frac{1}{2\sqrt{{{e}^{x}}}}{{e}^{x}}=\frac{1}{4}{{e}^{x/2}}\cot ({{e}^{x/2}})\]
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