d dx ( x^ _e x ) = - Vidyadip

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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