If $f$ is uniformly continuous on $[a,\infty)$,and $f’$ has definition on $[a,\infty)$,can one deduce that $f’$ is bounded?

If $ f$ is uniformly continuous on $ [a,\infty)$ ,and $ f’$ has definition on $ [a,\infty)$ ,can one deduce that $ f’$ is bounded on $ [a,\infty)$ ?

I know some functions like $ \sqrt{x}$ which has definition on the open set $ (0,\infty)$ , but which derivative is not bounded, so I wish the boundedness will be hold on a closed set $ [a,\infty)$