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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $y = {x^3}\log {\log _e}(1 + x)$, then $y''\,(0)$ equals
  • ✓
    $0$
  • B
    $-1$
  • C
    $6\,\,\log {_e}\,2$
  • D
    $6$
✓

Answer

Correct option: A.
$0$
a
(a) $y = {x^3}\log {\log _e}(1 + x)$

==> $y' = 3{x^2}\log {\log _e}\,(1 + x) + \frac{{{x^3}}}{{1 + x}}.\frac{1}{{{{\log }_e}(1 + x)}}$

==> $y'' = 6x\log {\log _e}(1 + x) + \frac{{3{x^2}}}{{{{\log }_e}(1 + x)}}.\frac{1}{{(1 + x)}}$

$ - \frac{{{x^3}}}{{{{(1 + x)}^2}{{\log }_e}(1 + x)}} - \frac{{{x^3}}}{{{{(1 + x)}^2}}}.\frac{1}{{{{[{{\log }_e}(1 + x)]}^2}}} + \frac{{3{x^2}}}{{(1 + x){{\log }_e}(1 + x)}}$

==> $y''(0) = 0$.

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