Question

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

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

Answer 1

Answer: (D)

$\log y=\frac{x}{\log x}\log 2$
$\frac{dy}{dx}=\frac{\log x-x\times \frac{1}{x}}{(\log x{)}^2}\times \log 2=\frac{\log x-1}{(\log x{)}^2}\times \log 2$
${\Rightarrow \frac{dy}{dx}]}_{x=e}=\frac{\log e-1}{(\log e{)}^2}\times \log 2=\frac{1-1}{(\log e{)}^2}\times \log 2=0$