Using Brzozowski's derivatives method to construct a minimal DFA
The Brzozowski derivative of a language $L$ with respect to a symbol $a$ consists of all words $w$ such that $aw \in L$.

In your case, $L$ is the language generated by the regular expression $X$. Instead of solving your exact exercise, I will show how to compute the other Brzozowski derivative, with respect to $b$. A word generated by $(aa+b)^*ab(bb)^*$ has the form
$$ x_1 \ldots x_n ab (bb)^{2m}, $$
where $n \geq 0$, $x_i \in \{aa,b\}$, and $m \geq 0$. The only way that this can start with $b$ is if $n \geq 1$ and $x_1 = b$. Taking away $b$, we are left with
$$ x_2 \ldots x_n an (bb)^{2m}, $$
a collection of words which is described by the regular expression $(aa+b)^*ab(bb)^*$, the same one we started with.

Computing the derivative with respect to $a$ is slightly more complicated since there are two different cases, but I'll leave you to ponder that.

You might have also learned some rules for computing the derivative. In that case, you can compute the derivative by applying them mechanically.