A) \[\frac{1}{x(\text{In}\,\text{5)(In}\,\text{7)(lo}{{\text{g}}_{\text{7}}}x)}\]
B) \[\frac{1}{x(\text{ln}\,\text{5)(ln}\,\text{7)}}\]
C) \[\frac{1}{x(In\,x)}\]
D) None of these
Correct Answer: A
Solution :
\[f(x)={{\log }_{5}}({{\log }_{7}}x)\] Þ \[f(x)={{\log }_{5}}\left( \frac{{{\log }_{e}}x}{{{\log }_{e}}7} \right)\] Þ \[f(x)={{\log }_{5}}{{\log }_{e}}x-{{\log }_{5}}{{\log }_{e}}7\] Þ \[f(x)=\frac{{{\log }_{e}}{{\log }_{e}}x}{{{\log }_{e}}5}-{{\log }_{5}}{{\log }_{e}}7\] \[{f}'(x)=\frac{1}{x{{\log }_{e}}x\log 5}-0\] \[{f}'(x)=\frac{1}{x{{\log }_{e}}x\frac{{{\log }_{e}}5}{{{\log }_{e}}7}{{\log }_{e}}7}\]\[=\frac{1}{x(\ln 5)(\ln 7)({{\log }_{7}}x)}\].
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