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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
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
    $\frac{1}{x(\rm{In}\,x)}$
  • D
    None of these
✓

Answer

Correct option: A.
${1 \over {x({\rm{In}}\,{\rm{5)(In}}\,{\rm{7)(lo}}{{\rm{g}}_{\rm{7}}}x)}}$
a
(a) $f(x) = {\log _5}({\log _7}x)$ 

==> $f(x) = {\log _5}\left( {\frac{{{{\log }_e}x}}{{{{\log }_e}7}}} \right)$

==> $f(x) = {\log _5}{\log _e}x - {\log _5}{\log _e}7$

==> $f(x) = \frac{{{{\log }_e}{{\log }_e}x}}{{{{\log }_e}5}} - {\log _5}{\log _e}7$

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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