https://r2.erweima.ai/mathbot/image/1724661163320oz4inzw0.jpg

https://r2.erweima.ai/mathbot/image/1724661163321epgmvsjo.jpg

To solve the problem, let's break it down:

1. **Problem Statement**: The function is given as \( f(x) = 4x - 5 \), and you are asked to find \( f'(2) \).

2. **Derivative Definition**: The second image provides the definition of a derivative:
   \[
   f'(x) = \lim_{z \to x} \frac{f(z) - f(x)}{z - x}
   \]
   This definition can be used to find the derivative of \( f(x) \).

3. **Derivative Calculation**: For the function \( f(x) = 4x - 5 \), the derivative \( f'(x) \) is:
   \[
   f'(x) = \frac{d}{dx}(4x - 5)
   \]

   The derivative of \( 4x \) with respect to \( x \) is 4, and the derivative of a constant (like -5) is 0. So, we have:
   \[
   f'(x) = 4
   \]

4. **Evaluate at \( x = 2 \)**: Since \( f'(x) = 4 \) is constant, it remains the same for any value of \( x \). Thus:
   \[
   f'(2) = 4
   \]

**Conclusion**: The value of \( f'(2) \) is \( 4 \).

Would you like more details or explanations?

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Here are 5 questions related to the content discussed:
1. What is the geometric interpretation of the derivative \( f'(x) \)?
2. How does the derivative differ for nonlinear functions, like \( f(x) = x^2 \)?
3. Can the derivative \( f'(x) \) ever be negative? What does that indicate?
4. What is the relationship between the derivative and the slope of the tangent line to the curve at a given point?
5. How does the limit definition of the derivative apply when \( f(x) \) is not linear?

**Tip**: Understanding the definition of a derivative as a limit can help clarify why the derivative measures the rate of change of a function at a specific point.

Learn how to calculate the derivative f'(2) of the function f(x) = 4x - 5. This problem involves applying the definition of derivatives and understanding the derivative of a constant function. Suitable for Grades 11-12 students.

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