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

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
${d \over {dx}}({x^{{{\log }_e}x}}) = $
  • ✓
    $2{x^{({{\log }_e}x - 1)}}.{\log _e}x$
  • B
    ${x^{({{\log }_e}x - 1)}}$
  • C
    ${2 \over x}{\log _e}x$
  • D
    ${x^{({{\log }_e}x - 1)}}.{\log _e}x$
✓

Answer

Correct option: A.
$2{x^{({{\log }_e}x - 1)}}.{\log _e}x$
a
(a) Let $y = {x^{{{\log }_e}x}}$

==> ${\log _e}y = {\log _e}x{\log _e}x = {({\log _e}x)^2}$

==> $\frac{1}{y}\frac{{dy}}{{dx}} = 2{\log _e}x.\frac{1}{x}$

$\therefore \frac{{dy}}{{dx}} = 2{x^{({{\log }_e}x - 1)}}{\log _e}x$.

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