Show that ${\sqrt{x+1}} - {\sqrt x} \leq {\frac{1}{2\sqrt x}}$ for all $x \in (0, \infty)$.
I know that the derivative of ${\sqrt x} + 1$ is equal to ${\frac{1}{2\sqrt x}}$. But I don't know if there is another function I should start off with to help me figure out how this inequality works.