A) \[{{x}^{\sqrt{x}}}\frac{2+\log x}{2\sqrt{x}}\]
B) \[{{x}^{\sqrt{x}}}\frac{2+\log x}{\sqrt{x}}\]
C) \[\frac{2+\log x}{2\sqrt{x}}\]
D) None of these
Correct Answer: A
Solution :
\[y={{x}^{\sqrt{x}}}\Rightarrow {{\log }_{e}}y=\sqrt{x}\log x\] Þ \[\frac{1}{y}\frac{dy}{dx}=\sqrt{x}\frac{1}{x}+\frac{1}{2\sqrt{x}}\log x\] or \[\frac{dy}{dx}={{x}^{\sqrt{x}}}\left[ \frac{2+{{\log }_{e}}x}{2\sqrt{x}} \right]\]
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