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The given problem is:

\[ f(x) = x^2 + 8x \]
We are asked to find the derivative of \( f(x) \) at \( x = 4 \), i.e., \( f'(4) \).

### Step 1: Find the derivative \( f'(x) \).
To find the derivative, we'll differentiate \( f(x) \) with respect to \( x \).

\[
f'(x) = \frac{d}{dx}(x^2 + 8x) = 2x + 8
\]

### Step 2: Calculate \( f'(4) \).
Now, substitute \( x = 4 \) into \( f'(x) \).

\[
f'(4) = 2(4) + 8 = 8 + 8 = 16
\]

So, the value of \( f'(4) \) is **16**.

Would you like more details or have any questions?

Here are some related questions:

1. What is the general formula for the derivative of a polynomial function?
2. How do you find the derivative of a function using the power rule?
3. What is the significance of the derivative in real-world applications?
4. How do higher-order derivatives relate to the original function?
5. What does the derivative tell us about the behavior of the function at a given point?

**Tip:** Always double-check your differentiation steps to ensure the accuracy of your derivative calculations.

Learn how to find the derivative of the function f(x) = x^2 + 8x and evaluate it at x = 4. This problem involves key calculus concepts such as the power rule of differentiation. Suitable for students in Grades 9-12.

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