As the topics. Proving that $\partial P'\subset \partial P $ if and only in $P'\cap P^0 \subset P'^0$. I am not sure how to start
One way to define $\partial P$ is as $P'\setminus P^0$. So if $P'\cap P^0\subset P'^0,$ then we want to show $\partial P' \cap P^0$ is empty. But $\partial P'\subset P'$ since derived sets are closed, so that $\partial P'\cap P^0\subset P'\cap P^0\subset P'^0$, while by the definition of boundary $\partial P'\cap P'^0$ is empty.
Approach the converse via the contrapositive: assume $x\in P'\cap P^0 \setminus P'^0$.Then in particular $x\in P'\setminus P'^0,$ that is $x\in \partial P'$, while on the other hand $x\notin \partial P$ since it's in the interior of $P$. Thus $\partial P'$ is not a subset of $\partial P$.