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.