MathNet 2023-part-03 — 191 problems (EN–VI) Question 157

Determine all differentiable functions $F:(0 ;+\infty) \rightarrow \mathbb{R}$ for which: $F(1)=1$ and $F\left(\frac{1}{x}\right) F^{\prime}(x)=\frac{1}{x} \ln x$.