HINTS: Parts (a), (c), and (e) are very easy: each of them is just a matter of using the definition. For instance, to prove (a), you must show that if $x\in C'$, then $x\in D\,'$. Suppose that $x\in C'$; then by definition for each $\delta>0$ there is some $c\in C$ such that $0. But $C\subseteq D$, so $c\in D$. Thus, for each $\delta>0$ there is some $c\in D$ such that $0, which by definition means that $x\in D\,'$.
For (c) you need to recall that $x\in\overline A$ if and only if for each $\delta>0$ there is an $a\in A$ such that $d(x,a)<\delta$. Suppose that $x\in A\cup A'$, and let $\delta>0$. If $x\in A$, you can take $a$ to be $x$ itself, since $d(x,x)=0<\delta$, and if $x\in A'$, then there is an $a\in A$ such that $0, which certainly implies that $d(x,a)<\delta$!
For (e), let $B=\{b_1,\dots,b_n\}$ be a finite set, and let $x\in X$. What happens if you set $\delta=\min\{d(x,b_k):k=1,\dots,n\text{ and }x\ne b_k\}\;?$ Is $\delta>0$? Can you find a $b_k\in B$ such that $0?
Once you have (c), (d) is very easy; just remember that if $A$ is closed, then $\overline A=A$.
Half of (b) follows immediately from (a): $A\subseteq A\cup B$, so $A'\subseteq(A\cup B)'$, and similarly $B\,'\subseteq(A\cup B)'$, so $A'\cup B\,'\subseteq(A\cup B)'$. Thus, it only remains to show that $(A\cup B)'\subseteq A'\cup B\,'$. To prove this, let $x\in(A\cup B)'$, and show that $x\in A'\cup B\,'$, i.e., that either $x\in A'$ or $x\in B\,'$. The easiest way is probably to suppose that $x\notin A'$ and $x\notin B\,'$ and derive a contradiction by showing that $x\notin(A\cup B)'$ after all. Note: The definition of $A'$ immediately tells you that if $x\notin A'$, then there is some $\delta>0$ such that no $a\in A$ satifies the inequality $0: every point of $A$ is either equal to $x$ or at least $\delta$ distant from $x$.
The second sentence in (f) is a hint: apply (b) to $B\cup(A\setminus B)$. You’ll also want to use (e) here.
That’s quite a bit to work on, so I’m going to stop here. Part (g) is harder than the rest, but that’s why it comes with a very extensive hint; perhaps you’ll be ready to tackle it on your own after you’ve dealt with the first six parts.