We have, $x_y = \log x$ ...(i)
Taking log on both sides of (i), we get
$y \log x =\log \left(\log x\right)\Rightarrow y =\frac{\log \left(\log x\right)}{\log x} $ ....(ii) $ \therefore \:\frac{dy}{dx} = \frac{\log x\left(\frac{1}{\log x}\right)\left(\frac{1}{x}\right)-\log \left(\log x\right)\left(\frac{1}{x}\right) -\log \left(\log x\right)\left(\frac{1}{x}\right)}{\left(\log x\right)^{2}}$
The point where the curve cuts the x-axis is (e, 0).
$ \therefore \:\: \frac{dy}{dx}|_{at (e,0)} = \frac{1.1 . \frac{1}{e} - 0}{(1)^2} = \frac{1}{e}$