6. For $x>0$, let $f(x)=x^{x}$. Find all values of $x$ for which $f(x)=f^{\prime}(x)$.
Solution: $\square$
Let $g(x)=\log f(x)=x \log x$. Then $f^{\prime}(x) / f(x)=g^{\prime}(x)=1+\log x$. Therefore $f(x)=f^{\prime}(x)$ when $1+\log x=1$, that is, when $x=1$.