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 $x^2 - 1$.

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.