To differentiate the function \( f(x) = \frac{x}{(x+2)^2} \), we will use the **quotient rule**.

The quotient rule states:
\[
\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
where \( u(x) = x \) and \( v(x) = (x+2)^2 \).

### Step 1: Differentiate the numerator
\[
u(x) = x \quad \Rightarrow \quad u'(x) = 1
\]

### Step 2: Differentiate the denominator
\[
v(x) = (x+2)^2 \quad \Rightarrow \quad v'(x) = 2(x+2)
\]

### Step 3: Apply the quotient rule
\[
f'(x) = \frac{1 \cdot (x+2)^2 - x \cdot 2(x+2)}{((x+2)^2)^2}
\]

### Step 4: Simplify the expression
Now simplify the numerator:
\[
f'(x) = \frac{(x+2)^2 - 2x(x+2)}{(x+2)^4}
\]
Expand both terms in the numerator:
\[
(x+2)^2 = x^2 + 4x + 4
\]
\[
2x(x+2) = 2x^2 + 4x
\]

So, the numerator becomes:
\[
x^2 + 4x + 4 - (2x^2 + 4x) = -x^2 + 4
\]

Thus, the derivative simplifies to:
\[
f'(x) = \frac{-x^2 + 4}{(x+2)^4}
\]

### Final answer:
\[
f'(x) = \frac{-x^2 + 4}{(x+2)^4}
\]

Would you like more details or have any questions? Here are some related questions:

1. How do we apply the quotient rule in more complex scenarios?
2. What happens when the denominator has higher powers?
3. Can the derivative of a similar function be calculated using logarithmic differentiation?
4. How does chain rule apply in problems involving higher powers?
5. What are some real-life applications of derivatives in such rational functions?

**Tip:** Always simplify the expression after applying the quotient rule to avoid errors in further calculations.

Explore how to differentiate the function x/(x+2)^2 using the quotient rule. This step-by-step solution explains how to apply the quotient rule, differentiate the numerator and denominator, and simplify the resulting expression. Perfect for students in advanced high school calculus or college-level calculus.

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