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Let’s analyze **Exercise 58** to determine where the function \( y \) is differentiable. The piecewise function is defined as:

\[
y = \begin{cases} 
x^3 + 3 & \text{if } x < 0, \\
x^3 - 3 & \text{if } x \geq 0.
\end{cases}
\]

### Steps to Determine Differentiability
1. **Continuity Check**: For the function to be differentiable, it must first be continuous. Let us verify if \( y \) is continuous at \( x = 0 \).
   - From the left-hand side (\( x < 0 \)): \( \lim_{x \to 0^-} (x^3 + 3) = 0^3 + 3 = 3 \).
   - From the right-hand side (\( x \geq 0 \)): \( \lim_{x \to 0^+} (x^3 - 3) = 0^3 - 3 = -3 \).

   Since the left and right limits at \( x = 0 \) are not equal, the function is **not continuous** at \( x = 0 \). Therefore, \( y \) is **not differentiable** at \( x = 0 \).

2. **Behavior Elsewhere**: For all \( x \neq 0 \), each piece of the function (\( x^3 + 3 \) and \( x^3 - 3 \)) is a polynomial, and polynomials are differentiable everywhere. Thus, \( y \) is differentiable for \( x \neq 0 \).

### Conclusion
The function is differentiable everywhere except at \( x = 0 \), where it is neither continuous nor differentiable.

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Would you like more details or assistance with other exercises? Here are some related questions to expand your understanding:

1. How do you test for differentiability at a point using derivatives?
2. What is the relationship between continuity and differentiability?
3. How does a piecewise function affect continuity and differentiability?
4. Can a function be differentiable but not continuous? Why or why not?
5. What role does the slope (derivative) from the left and right play in differentiability?

**Tip**: Always check for continuity first when determining differentiability, as it is a necessary (but not sufficient) condition.

Determine the x-values where the function in Exercise 58 is differentiable and explain your reasoning.

Explore the differentiability of a piecewise function defined as y = x^3 + 3 for x < 0 and y = x^3 - 3 for x ≥ 0. Learn how continuity and limits determine differentiability, with step-by-step reasoning.

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