The image contains two tasks related to proving differentiability theorems for functions involving powers:

1. First Task:
   Prove the differentiability theorem for $x^a$, where $a$ is a negative integer. The proof should be based on the theorem for differentiability of $x^n$, where $n$ is a natural number.

2. Second Task:
   Present and prove a theorem about the differentiability of $x^t$, where $t$ is half of an integer (e.g., $t = b - \frac{1}{2}$, where $b$ is an integer). You are allowed to use the differentiability theorem for $\sqrt{x}$ without proof, and you must justify that $b - \frac{1}{2}$ is not an integer.

Would you like me to go step by step on either or both of these tasks? Let me know how detailed you want the explanation.

Here are 5 questions related to this topic to expand understanding:

1. What is the general proof structure for the differentiability of $x^n$ for natural $n$?
2. Why does the differentiability of $x^n$ extend to $x^a$ for negative integers?
3. How does the differentiability of $\sqrt{x}$ support the proof for $x^t$ when $t = b - \frac{1}{2}$?
4. What is the relationship between differentiability and continuity in these proofs?
5. Can these differentiability proofs be generalized to other fractional or irrational powers?

Tip: Start by writing the derivative definition explicitly using limits; this often simplifies the transition between different powers.