https://file.aiquickdraw.com/mathbot/file/1726090221639gtnb6nto.jpg

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 \(nx^{n-1}\).

Using the definition, calculate the derivative of the function. Then find the values of the derivative as specified. f(x) = 8 + x^2; f'(-3), f'(0), f'(7)

Learn how to calculate the derivative of the function f(x) = 8 + x^2 and evaluate the derivative at points x = -3, x = 0, and x = 7. This problem covers basic differentiation concepts, including polynomial differentiation rules. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation