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.