It's known that for a given group $G$ and the derived subset $[G,G]$ it's not true that every element of $[G,G]$ is a commutator. I've seen a condition that shows when there are elements in $[G,G]$ which are not commutator.
For any group $G$ with $$\left|\left[G:Z\left(G\right)\right]\right|^{2}<\left|\left[G,G\right]\right|$$ there are elements in $[G,G]$ that are not commutator.
I have not seen any proof of this, it would be really nice if someone proof that or give me a link about the problem.
Since $[gk_1,gk_2] = [g,h]$ for all $k_1,k_2 \in Z(G)$, there are at most $[G:Z(G)]^2$ distinct commutators.
So if $[G:Z(G)]^2 < |[G,G]|$ then $|[G,G]|$ is larger than the number of commutators, so there must be elements of $[G,G]$ that are not commutators.