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

Let’s break this down.

### 1. Function Simplification:
The function is given as:

\[
f(x) = x^2 + 2x^2 + 6
\]

First, combine the terms involving \( x^2 \):

\[
f(x) = 3x^2 + 6
\]

### 2. Derivative using the Alternative Form:
The alternative definition of a derivative at a point \( c \) is:

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

Here, \( c = -2 \). Now, we'll follow these steps:

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

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

#### Step 3: Substitute into the limit definition
Now, substitute \( f(c+h) \) and \( f(c) \) into the limit definition:

\[
f'(-2) = \lim_{{h \to 0}} \frac{(18 - 12h + 3h^2) - 18}{h}
\]

Simplify:

\[
f'(-2) = \lim_{{h \to 0}} \frac{-12h + 3h^2}{h} = \lim_{{h \to 0}} (-12 + 3h)
\]

#### Step 4: Take the limit as \( h \to 0 \)
\[
f'(-2) = -12
\]

Thus, the derivative at \( x = -2 \) is:

\[
f'(-2) = -12
\]

Would you like any additional details on this solution or further clarifications?

Here are five related questions for practice:
1. Find the derivative of \( f(x) = 4x^2 + 5x - 3 \) at \( x = 1 \).
2. Use the alternative definition of the derivative to find \( f'(1) \) for \( f(x) = x^3 - 2x + 1 \).
3. Determine if the function \( f(x) = |x| \) is differentiable at \( x = 0 \) using the limit definition of the derivative.
4. Calculate the derivative of \( f(x) = \sin(x) + x^2 \) at \( x = \pi \) using the alternative form of the derivative.
5. What is the geometrical interpretation of the derivative of a function at a given point?

**Tip**: When using the alternative definition of the derivative, always simplify the function expressions before applying the limit to avoid unnecessary complexity.

Use the alternative form of the derivative to find the derivative at x = c (if it exists). If the derivative does not exist at c, enter UNDEFINED. The function is f(x) = x^2 + 2x^2 + 6, with c = -2.

Explore how to find the derivative of the function f(x) = 3x^2 + 6 at x = -2 using the limit definition of the derivative. This guide provides step-by-step calculations to understand derivatives in calculus, suitable for college-level students.

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