We are given the function $f(x) = -8x + 4$ and asked to find its derivative $f'(x)$, and evaluate $f'(2)$, $f'(0)$, and $f'(-6)$.

Step 1: Find $f'(x)$

The function $f(x) = -8x + 4$ is a linear function, so its derivative is the slope of the function.

The derivative of $f(x) = -8x + 4$ is: $f'(x) = \frac{d}{dx}(-8x + 4) = -8$

Step 2: Evaluate $f'(2)$, $f'(0)$, and $f'(-6)$

Since $f'(x) = -8$, it remains constant for all values of $x$. Therefore:

$f'(2) = -8$ $f'(0) = -8$ $f'(-6) = -8$

Final Answer:

• $f'(x) = -8$
• $f'(2) = -8$
• $f'(0) = -8$
• $f'(-6) = -8$

Would you like further details or have any other questions?

Here are some related questions to explore:

1. What is the derivative of a constant function?
2. How does the derivative of a linear function compare to a quadratic function?
3. Can you explain why the derivative of a linear function is constant?
4. How would the derivative change if the function had a higher-degree term, such as $x^2$?
5. What are the practical applications of finding derivatives of linear functions?

Tip: For linear functions, the derivative is always the slope of the line.