How can I prove that this function is Lipschitz?

Question

I am stuck with the following problem. I have two function $f(x)=\frac{\sqrt{1+x^2}}{(1+|x|)}$ and $g(x)=\frac{x}{(1+|x|)^2}$. Are they Lipschitz (on the whole real line) ? I think so, but I can't find the constant $L$. Instead of trying to compute $L$, I also thought about computing the derivatives of $f$ and $g$ and see whether they are bounded or not. In case they are, we could conclude. Is it right ? Thank you very much for any help you can provide ? :)