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The image contains a mathematical problem involving the derivative of a given function, along with some handwritten notes. Let's break it down:

### Problem Statement:
Given the function \( f(x) = \frac{x^3}{3} - x \), find \( \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \).

This expression represents the definition of the derivative of \( f(x) \) at any point \( x \).

### Steps to Solve:
1. **Substitute \( f(x+h) \)**:
   \[
   f(x+h) = \frac{(x+h)^3}{3} - (x+h)
   \]
   
2. **Expand \( (x+h)^3 \)**:
   \[
   (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3
   \]
   Therefore:
   \[
   f(x+h) = \frac{x^3 + 3x^2h + 3xh^2 + h^3}{3} - (x+h)
   \]
   
3. **Simplify**:
   \[
   f(x+h) = \frac{x^3}{3} + x^2h + \frac{xh^2}{3} + \frac{h^3}{3} - x - h
   \]
   
4. **Compute \( f(x+h) - f(x) \)**:
   \[
   f(x+h) - f(x) = \left(\frac{x^3}{3} + x^2h + \frac{xh^2}{3} + \frac{h^3}{3} - x - h\right) - \left(\frac{x^3}{3} - x\right)
   \]
   Simplifying further:
   \[
   f(x+h) - f(x) = x^2h + \frac{xh^2}{3} + \frac{h^3}{3} - h
   \]
   
5. **Divide by \( h \)**:
   \[
   \frac{f(x+h) - f(x)}{h} = x^2 + \frac{xh^2}{3h} + \frac{h^3}{3h} - 1
   \]
   Simplify:
   \[
   \frac{f(x+h) - f(x)}{h} = x^2 + \frac{xh}{3} + \frac{h^2}{3} - 1
   \]
   
6. **Take the limit as \( h \to 0 \)**:
   \[
   \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} = x^2 - 1
   \]

Thus, the derivative of \( f(x) \) is \( f'(x) = x^2 - 1 \).

### Interpretation:
This means the rate of change of the function \( f(x) = \frac{x^3}{3} - x \) with respect to \( x \) is \( x^2 - 1 \).

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Would you like more details on this problem or have any other questions? Here are 5 questions that expand on this information:

1. What is the geometric interpretation of the derivative \( f'(x) \) for the given function?
2. How would you find the points where the derivative \( f'(x) = 0 \)?
3. How does the derivative relate to the concavity of the function \( f(x) \)?
4. What would the second derivative \( f''(x) \) tell you about the function's behavior?
5. How would you apply this derivative in a real-world problem involving rates of change?

**Tip:** Understanding the derivative as a limit is crucial, as it forms the foundation for more complex calculus concepts like integration and differential equations.

Find the limit \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \) for the function \( f(x) = \frac{x^3}{3} - x \).

Explore how to find the limit \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \) for the function \( f(x) = \frac{x^3}{3} - x \). This problem covers key calculus concepts, including the definition of a derivative and the use of limits. Suitable for advanced high school or early college students.

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