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To find the instantaneous rate of change of the function \( f(x) = x^2 + 4x \) at \( x = 6 \), we need to compute the derivative of the function and then evaluate it at \( x = 6 \).

### Step 1: Find the derivative
Given \( f(x) = x^2 + 4x \), we differentiate it with respect to \( x \).

\[
f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(4x)
\]

\[
f'(x) = 2x + 4
\]

### Step 2: Evaluate the derivative at \( x = 6 \)
Now substitute \( x = 6 \) into \( f'(x) \):

\[
f'(6) = 2(6) + 4 = 12 + 4 = 16
\]

Thus, the instantaneous rate of change of \( f(x) \) at \( x = 6 \) is \( 16 \).

### Answer:
The correct choice is **C) 16**.

Would you like more details on derivatives or have any questions?

Here are 5 related questions to further your understanding:
1. How do you compute the derivative of a polynomial function in general?
2. What is the physical meaning of an instantaneous rate of change in practical terms?
3. How would the answer change if the function had a cubic term?
4. What is the difference between the average rate of change and the instantaneous rate of change?
5. Can you find the second derivative of the function \( f(x) = x^2 + 4x \)?

**Tip**: Derivatives give you the slope of the tangent line at any given point on a curve!

Find the instantaneous rate of change for the function f(x) = x^2 + 4x at x = 6.

Find the instantaneous rate of change of the function f(x) = x^2 + 4x at x = 6. This problem involves using derivatives and is suitable for students in Grades 10-12 studying calculus. Learn how to compute the derivative and apply it to solve real-world problems.

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