MCQ
Derivative of the function $f(x) = {\log _5}({\log _7}x)$, $x > 7$ is
- ✓${1 \over {x({\rm{In}}\,{\rm{5)(In}}\,{\rm{7)(lo}}{{\rm{g}}_{\rm{7}}}x)}}$
- B${1 \over {x({\rm{ln}}\,{\rm{5)(ln}}\,{\rm{7)}}}}$
- C$\frac{1}{x(\rm{In}\,x)}$
- DNone of these
==> $f(x) = {\log _5}\left( {\frac{{{{\log }_e}x}}{{{{\log }_e}7}}} \right)$
==> $f(x) = {\log _5}{\log _e}x - {\log _5}{\log _e}7$
==> $f(x) = \frac{{{{\log }_e}{{\log }_e}x}}{{{{\log }_e}5}} - {\log _5}{\log _e}7$
Now, $f'(x) = \frac{1}{{x{{\log }_e}x\log 5}} - 0$
==> $f'(x) = \frac{1}{{x{{\log }_e}x\frac{{{{\log }_e}5}}{{{{\log }_e}7}}{{\log }_e}7}}$
$ = \frac{1}{{x(\ln 5)(\ln 7)({{\log }_7}x)}}$.
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If $f$ is continuous, then which of the following hold$(s)$ for all $n$ ?
$(A)$ $a_{n-1}-b_{n-1}=0$ $(B)$ $a_n-b_n=1$ $(C)$ $a_n-b_{n+1}=1$ $(D)$ $a_{n-1}-b_n=-1$
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