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You need to find the derivative of the function \( f(x) = x^3 + 2x^2 + 6 \) at \( x = -2 \).

To solve this, we first differentiate \( f(x) \) with respect to \( x \). The derivative \( f'(x) \) is found by differentiating each term of the function separately:

1. The derivative of \( x^3 \) is \( 3x^2 \).
2. The derivative of \( 2x^2 \) is \( 4x \).
3. The derivative of the constant \( 6 \) is \( 0 \) (since the derivative of any constant is zero).

Thus, the derivative \( f'(x) \) is:
\[ f'(x) = 3x^2 + 4x \]

Next, we substitute \( x = -2 \) into \( f'(x) \) to find \( f'(-2) \):
\[ f'(-2) = 3(-2)^2 + 4(-2) \]
\[ f'(-2) = 3 \cdot 4 - 8 \]
\[ f'(-2) = 12 - 8 \]
\[ f'(-2) = 4 \]

Therefore, \( f'(-2) = 4 \).

Would you like to learn more about this process or have any other questions?

Here are five related questions you might find helpful:
1. What is the derivative of a polynomial of higher degree, such as \( x^5 + 3x^3 - 10x + 7 \)?
2. How do you calculate higher-order derivatives for a function like \( x^3 + 2x^2 + 6 \)?
3. What methods can be used to find derivatives other than straightforward differentiation?
4. How does the derivative help in understanding the behavior of a function?
5. What is the derivative of the function \( f(x) = x^2 - 4x + 4 \) and at what points is this function's slope zero?

**Tip:** To better understand the behavior of a function, graph its derivative alongside the original function. This can help visualize how the slope changes across different values of \( x \).

Use the alternative form of the derivative to find the derivative at x = c (if it exists). f(x) = x^3 + 2x^2 + 6, c = -2.

Find the derivative of the function f(x) = x^3 + 2x^2 + 6 at x = -2 using the alternative form. Learn how to apply the power rule of differentiation and calculate the slope of the function at a specific point.

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