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The problem asks to find the derivative of the function \( f(x) = x^2 + 2x^2 + 6 \) at \( x = -2 \), using the alternative form of the derivative.

Let's break this down step-by-step.

### Function:
\[
f(x) = x^2 + 2x^2 + 6
\]
Simplifying the terms:
\[
f(x) = 3x^2 + 6
\]

### The Alternative Form of the Derivative:
The alternative definition of a derivative at \( x = c \) is given by:
\[
f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}
\]
Here, \( c = -2 \).

#### Step 1: Calculate \( f(c) = f(-2) \)
\[
f(-2) = 3(-2)^2 + 6 = 3(4) + 6 = 12 + 6 = 18
\]

#### Step 2: Set up the limit definition of the derivative
We need to evaluate the following limit:
\[
f'(-2) = \lim_{x \to -2} \frac{f(x) - f(-2)}{x + 2}
\]
This becomes:
\[
f'(-2) = \lim_{x \to -2} \frac{3x^2 + 6 - 18}{x + 2} = \lim_{x \to -2} \frac{3x^2 - 12}{x + 2}
\]

#### Step 3: Simplify the expression
\[
f'(-2) = \lim_{x \to -2} \frac{3(x^2 - 4)}{x + 2}
\]
Notice that \( x^2 - 4 \) is a difference of squares, so we can factor it:
\[
f'(-2) = \lim_{x \to -2} \frac{3(x - 2)(x + 2)}{x + 2}
\]

Now, cancel out the \( x + 2 \) terms:
\[
f'(-2) = \lim_{x \to -2} 3(x - 2)
\]

#### Step 4: Evaluate the limit
Substitute \( x = -2 \):
\[
f'(-2) = 3(-2 - 2) = 3(-4) = -12
\]

### Final Answer:
\[
f'(-2) = -12
\]

Let me know if you'd like further clarification or details on any step! Here are some follow-up questions to consider:

1. How does the alternative form of the derivative differ from the basic limit definition?
2. What is the significance of factoring in simplifying limits in derivative calculations?
3. How can you handle cases where the derivative does not exist at a certain point?
4. Can you apply this derivative method to other types of functions, like rational functions?
5. What are the benefits of using the limit definition over other methods to find derivatives?

**Tip**: Always check if a function can be factored to simplify limits when using the definition of the derivative.

Use the alternative form of the derivative to find the derivative at x = c for f(x) = x^2 + 2x^2 + 6, c = -2.

Learn how to calculate the derivative of f(x) = 3x^2 + 6 at x = -2 using the alternative form of the derivative. This problem explores key calculus concepts including derivatives, limits, and algebraic simplification.

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