To solve this problem, we'll first calculate the derivative of the given function using the standard differentiation rules and then evaluate the derivative at the specified points $x = -3$, $x = 0$, and $x = 7$.

Step 1: Find the Derivative of $f(x) = 8 + x^2$

Using basic derivative rules:

• The derivative of a constant $8$ is $0$.
• The derivative of $x^2$ is $2x$.

Thus, the derivative of the function is: $f'(x) = 2x$

Step 2: Evaluate the Derivative at the Given Points

• $f'(-3) = 2(-3) = -6$
• $f'(0) = 2(0) = 0$
• $f'(7) = 2(7) = 14$

Summary of Results:

$f'(x) = 2x$

• $f'(-3) = -6$
• $f'(0) = 0$
• $f'(7) = 14$

Let me know if you'd like any further details or explanations.

Relative Questions:

1. What is the geometric interpretation of the derivative?
2. How does the derivative relate to the slope of the tangent line at a given point?
3. What are the basic rules for differentiating polynomial functions?
4. How would the result change if the function had been $f(x) = 8 + 3x^2$?
5. Can we calculate higher-order derivatives for this function?

Tip:

Always remember that the derivative of a constant is zero, and the derivative of $x^n$ is $nx^{n-1}$.