A) \[\frac{{{\log }_{2}}e}{x{{\log }_{e}}x}\]
B) \[\frac{1}{{{\log }_{e}}x{{\log }_{e}}2}\]
C) \[\frac{1}{{{\log }_{e}}{{(2x)}^{x}}}\]
D) None of these
Correct Answer: A
Solution :
\[y={{\log }_{2}}[{{\log }_{2}}(x)]={{\log }_{e}}({{\log }_{e}}x.{{\log }_{2}}e).{{\log }_{2}}e\] \[=[{{\log }_{e}}{{\log }_{e}}x+{{\log }_{e}}({{\log }_{2}}e)]{{\log }_{2}}e\] \[\therefore \frac{dy}{dx}={{\log }_{2}}e.\frac{1}{x{{\log }_{e}}x}\].
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