I'm looking for an unbounded function $f(x)$ such that $\forall x \in \mathbb{R}$, $$ \frac{\textrm{d} f}{\textrm{d} x} > 0\quad\textrm{and}\quad \frac{\textrm{d}^{2} f}{\textrm{d} x^{2}} < 0. $$ I was able to come up with $$ f(x) = \log\Big(x+\sqrt{x^{2}+1}\Big) - \Big[ x^{2} - x\sqrt{x^{2}+1} \Big], $$ but are there simpler such functions?
I found this question, but I don't want the function to be asymptotically linear.