https://file.aiquickdraw.com/mathbot/file/1727314996041edbo5emr.jpg

The problem in the image is asking you to:

1. **Find the derivative function \( f'(x) \)** of the given function \( f(x) = 6x^2 + 6x \) using limits.
2. **Evaluate** the derivative \( f'(a) \) for the values \( a = -2 \) and \( a = 4 \).

### Part (a): Derivative function \( f'(x) \)
The derivative of \( f(x) \) is the limit definition:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Given \( f(x) = 6x^2 + 6x \), let's calculate the derivative using the power rule:
- The derivative of \( 6x^2 \) is \( 12x \)
- The derivative of \( 6x \) is \( 6 \)

So, \( f'(x) = 12x + 6 \).

### Part (b): Evaluate \( f'(a) \) for \( a = -2 \) and \( a = 4 \)

1. For \( a = -2 \):
   \[
   f'(-2) = 12(-2) + 6 = -24 + 6 = -18
   \]

2. For \( a = 4 \):
   \[
   f'(4) = 12(4) + 6 = 48 + 6 = 54
   \]

### Final Answer:
- \( f'(x) = 12x + 6 \)
- \( f'(-2) = -18 \)
- \( f'(4) = 54 \)

Do you need further details or have any questions?

Here are 5 related questions to expand your knowledge:
1. How would the derivative change if \( f(x) \) had an additional constant term?
2. Can you apply the limit definition of a derivative to solve this problem step-by-step?
3. What is the graphical interpretation of the derivative \( f'(x) \)?
4. How do you find the second derivative of \( f(x) = 6x^2 + 6x \)?
5. How would the values of \( f'(a) \) change if \( a = 0 \)?

**Tip**: The derivative tells you the slope of the function at any point \( x \).

Use limits to find the derivative function f' for the following function f. f(x) = 6x^2 + 6x; a = -2, 4

Learn how to find the derivative of f(x) = 6x^2 + 6x using limits and evaluate the derivative at points a = -2 and a = 4. This step-by-step solution covers key calculus concepts and the application of the power rule for differentiation.

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