Thank you for the clarification. Let's go through the steps again with the correct function \( f(x) = 4x - x^2 + 2 \).

### Step 1: Find the derivative of \( f(x) \).
The function is:
\[
f(x) = 4x - x^2 + 2
\]
The derivative \( f'(x) \) with respect to \( x \) is:
\[
f'(x) = \frac{d}{dx}(4x) - \frac{d}{dx}(x^2) + \frac{d}{dx}(2)
\]

Let's differentiate each term:
- \( \frac{d}{dx}(4x) = 4 \)
- \( \frac{d}{dx}(x^2) = 2x \)
- \( \frac{d}{dx}(2) = 0 \) (since the derivative of a constant is zero)

So, the derivative \( f'(x) \) is:
\[
f'(x) = 4 - 2x
\]

### Step 2: Evaluate the derivative at \( x = -2 \).
Substitute \( x = -2 \) into \( f'(x) \):
\[
f'(-2) = 4 - 2(-2)
\]
\[
f'(-2) = 4 + 4 = 8
\]

### Final Answer:
The rate of change of the function at \( x = -2 \) is \( 8 \).

Would you like more details or have any questions?

Here are five related questions you might find useful:
1. How does changing the coefficients in a quadratic function affect its derivative?
2. What is the geometric interpretation of the rate of change for a quadratic function?
3. How do you determine the maximum or minimum points using the derivative?
4. How can you apply the concept of derivatives to real-world problems, such as velocity?
5. What is the significance of the sign of the derivative in different intervals of the function?

**Tip:** Always ensure that the correct form of the function is used before performing calculations to avoid errors in the final result.

Learn how to find the derivative of a quadratic function step-by-step. This problem involves differentiating the function f(x) = 4x - x^2 + 2 and evaluating it at x = -2. Gain insights into differentiation rules and their application in calculus.

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