First slide

Derivatives

Question

$If y = 2^{\frac{x}{\log x}} then (\frac{d y}{d x}) at x = e is$

Moderate

A

0
B

log2
C

e
D

$e^{2}$

Solution

$\log y = \frac{x}{\log x} \log 2$
$\begin{aligned} \frac{1}{y} \frac{d y}{d x} = \log 2 [\frac{\log x - x \frac{1}{x}}{(\log x)^{2}}] \\ \frac{d y}{d x} = \log 2 \cdot y \cdot \frac{(\log x - 1)}{(\log x)^{2}} \\ = \log 2 \cdot 2^{e} \cdot \frac{(1 - 1)}{(\log e)^{2}} = 0 \end{aligned}$

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