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The derivative of the given function \( f(x) = \frac{x^2 + 6x}{2x^2 + 4x + 4} \) is:

\[
f'(x) = \frac{(-4x - 4)(x^2 + 6x)}{(2x^2 + 4x + 4)^2} + \frac{2x + 6}{2x^2 + 4x + 4}
\]

Would you like a detailed step-by-step explanation of how this derivative was obtained?

Here are some related questions to explore this topic further:
1. How can you find the critical points of the function \( f(x) \)?
2. What is the behavior of \( f(x) \) as \( x \to \infty \)?
3. How do you simplify complex rational expressions after taking derivatives?
4. Can you determine whether the function \( f(x) \) has any inflection points?
5. How do you analyze the concavity of \( f(x) \) using the second derivative?

**Tip:** When finding the derivative of a quotient, use the quotient rule: \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\).

Find the derivative of the function f(x) = (x^2 + 6x) / (2x^2 + 4x + 4).

Learn how to compute the derivative of f(x) = (x^2 + 6x) / (2x^2 + 4x + 4) using the quotient rule. The solution includes step-by-step application of calculus concepts like derivatives and the quotient rule. Suitable for advanced high school or undergraduate calculus students.

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