Uniformly continuous and differentiable

I have this question. I know that a differentiable function is continuous and that a differentiable function with bounded first derivative is lipschits continuous. Now i wanted to know if a differentiable function is also uniformly continious. I know it’s not beceause a continous functions isn’t always uniformly continous but are there conditions so that a differentiable function is uniformly continous?

EDIT: now i know that the condition is the same: so the first derivative need to be bounded i tried to prove it but i’m stuck. I have a counter example for if the function isn’t bounded, namely f(x)=1/x but i’m stuck at proving when the condition is used. Can someone help me.