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!