Prove that if $G$ is a $p$-group, $p$ is prime, then $G'=[G,G] \subseteq Z(G)$

Question

I have proved if $G$ is a $p$-group, $G$ is nilpotent, and the lower central series and the upper central series have a finite length and both cover from $G$ to/downto $\lbrace 1 \rbrace$. But I do not know how to prove $[G,G]\subseteq Z(G)$. Some hint?

Answer 1

The claim of the title is not true in general, see here:

Very generic question about Commutator and Center

Every group $G$ with $G'\subseteq Z(G)$ is $2$-step solvable, hence metabelian. Not every $p$-group is metabelian.