Let y = xlo g _ e x . If the value of (dy)/(dx) at x = e ^ 4 is k ,...

$\dfrac{dy}{dx} = \dfrac{1}{2\sqrt{xlo{g}_{e}x}}\dfrac{d}{dx}\left[xlo{g}_{e}x\right]$
$= \dfrac{1}{2\sqrt{xlo{g}_{e}x}}\left[x \times \dfrac{1}{x} + 1 \times lo{g}_{e}x\right]$
$⇒ {\left[\dfrac{dy}{dx}\right]}_{x = {e}^{4}} = \dfrac{1}{2\sqrt{{e}^{4} \times 4}}\left(1 + 4\right) = \dfrac{5}{4{e}^{2}}\left(∵ lo{g}_{e}e = 1\right)$

Hence, $4{e}^{3}k = 4{e}^{3}\left(\dfrac{5}{4{e}^{2}}\right) = 5e = 13 . 5$