Answer
\[f''(e)=-e^{-3}\]
Work Step by Step
It is given that \[f(x)=\frac{\ln x}{x}\]
Differentiating $f(x)$ with respect to $x$ using quotient rule
\[f'(x)=\frac{\frac{1}{x}\cdot (x)-\ln x\cdot (1)}{x^2}\]
\[\Rightarrow f'(x)=\frac{1-\ln x}{x^2}\]
Differentiating $f'(x)$ with respect to $x$
\[f''(x)=\frac{\left(\frac{-1}{x}\right)x^2-(1-\ln x)2x}{x^4}\]
\[f''(x)=\frac{-x-2x+2x\ln x}{x^4}\]
\[\Rightarrow f''(x)=\frac{-3+2\ln x}{x^3}\]
\[\Rightarrow f''(e)=\frac{-3+2}{e^3}=-e^{-3}\]
Hence, \[f''(e)=-e^{-3}\]
