When this $f(x)=\frac{\log x}{x}$ decreasing and when is it increasing

Question

When this$f(x)=\frac{\log x}{x}$ decreasing and when is it increasing?

Usually in graph increasing function identified by $f'(x)>0$ and vice versa for decreasing. In this case, however, after I get $f'(x) =\frac{ 1 - \log x}{x^2}$ i am confused in deciding when is it increasing or decreasing.

Answer 1

$f'(x) >0$ if $x <e$ and $f'(x) <0$ if $x >e$. So $f$ is increasing in $(0,e)$ and decreasing in $(e,\infty)$.

Answer 2

The derivative with respect o $x$ of the above function is $$ f'(x) = x^{-2} \left( 1-\ln (x) \right) \, . $$

Thus for $x<e$ the function is increasing and for $x>e$ it is decreasing.