Question

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

• A. $e$
• B. $2^e \log \, 2 $
• C. $\log \, 2 $
• D. $0$

Answer 1

Answer: (D)

$\log y=\frac{x}{\log x} \log 2$
$\frac{d y}{d x}=\frac{\log x-x \times \frac{1}{x}}{(\log x)^{2}} \times \log 2=\frac{\log x-1}{(\log x)^{2}} \times \log 2$
$\left.\Rightarrow \frac{d y}{d x}\right]_{x=e}=\frac{\log e-1}{(\log e)^{2}} \times \log 2=\frac{1-1}{(\log e)^{2}} \times \log 2=0$