Hints for two proofs Please do not give me the answers. I only want hints to approach these two proof problems I'm struggling with:

*

*There exists some differentiable function $f(x)$ such that $f'(x)\ne f(x)$ but $f''(x)=f(x)$

*There exists some $x\in \mathbb R$ such that $x^2\in \mathbb \{R -\mathbb Q\}$ while $x^4\in \mathbb R$
For the first one I started out defining $f'$ as a limit but I couldn't come up with any way to utilize this for my proof. For the second one I've tried doing it by cases, considering $x<0, x=0, x>0$. But I didn't get anywhere doing this.

Hint
$1.$ Think about $\sin \alpha$;
$2.$ Think about $x^2=\sqrt{3}$.
