A) \[\frac{{{\log }_{2}}e}{{{\log }_{e}}x}\]
B) \[\frac{{{\log }_{2}}e}{x{{\log }_{x}}2}\]
C) \[\frac{{{\log }_{2}}x}{{{\log }_{e}}2}\]
D) \[\frac{{{\log }_{2}}e}{{{\log }_{2}}x}\]
E) \[\frac{{{\log }_{2}}e}{x{{\log }_{e}}x}\]
Correct Answer: E
Solution :
Given, \[y={{\log }_{2}}{{\log }_{2}}(x)\] \[=\frac{{{\log }_{e}}{{\log }_{2}}(x)}{{{\log }_{e}}2}\] \[=\frac{{{\log }_{e}}\left[ \frac{{{\log }_{e}}x}{{{\log }_{e}}2} \right]}{{{\log }_{e}}2}\] \[\Rightarrow \] \[y=\frac{{{\log }_{e}}{{\log }_{e}}x-{{\log }_{e}}{{\log }_{e}}2}{{{\log }_{e}}2}\] On differentiating w.r.t.\[x,\]we get \[\frac{dy}{dx}=\frac{1}{{{\log }_{e}}2}\left[ \frac{1}{x{{\log }_{e}}x}-0 \right]\] \[=\frac{{{\log }_{2}}e}{x{{\log }_{e}}x}\]
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