We are tasked with finding the second derivative \( f''(2) \) for the function:

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

### Step 1: First derivative \( f'(x) \)
We'll apply the quotient rule to differentiate \( f(x) \).

The quotient rule is:

\[
\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]

For \( f(x) = \frac{x^2}{4 + x} \), let:
- \( u(x) = x^2 \) with \( u'(x) = 2x \)
- \( v(x) = 4 + x \) with \( v'(x) = 1 \)

Using the quotient rule:

\[
f'(x) = \frac{(2x)(4 + x) - (x^2)(1)}{(4 + x)^2}
\]

Simplifying the numerator:

\[
f'(x) = \frac{2x(4 + x) - x^2}{(4 + x)^2} = \frac{8x + 2x^2 - x^2}{(4 + x)^2} = \frac{8x + x^2}{(4 + x)^2}
\]

### Step 2: Second derivative \( f''(x) \)
Now, we need to differentiate \( f'(x) = \frac{8x + x^2}{(4 + x)^2} \) using the quotient rule again. Let:
- \( u(x) = 8x + x^2 \) with \( u'(x) = 8 + 2x \)
- \( v(x) = (4 + x)^2 \) with \( v'(x) = 2(4 + x) \)

Applying the quotient rule:

\[
f''(x) = \frac{(8 + 2x)(4 + x)^2 - (8x + x^2) \cdot 2(4 + x)}{(4 + x)^4}
\]

Simplifying the numerator:

\[
f''(x) = \frac{(8 + 2x)(4 + x)^2 - 2(8x + x^2)(4 + x)}{(4 + x)^4}
\]

### Step 3: Evaluate at \( x = 2 \)

First, simplify the expression at \( x = 2 \):

\[
f'(2) = \frac{8(2) + 2^2}{(4 + 2)^2} = \frac{16 + 4}{6^2} = \frac{20}{36} = \frac{5}{9}
\]

Next, substitute \( x = 2 \) into the second derivative formula:

\[
f''(2) = \frac{(8 + 2(2))(4 + 2)^2 - 2(8(2) + 2^2)(4 + 2)}{(4 + 2)^4}
\]

Now, compute:

\[
f''(2) = \frac{(8 + 4)(6)^2 - 2(16 + 4)(6)}{6^4}
\]

\[
f''(2) = \frac{12(36) - 2(20)(6)}{6^4}
\]

\[
f''(2) = \frac{432 - 240}{6^4}
\]

\[
f''(2) = \frac{192}{1296} = \frac{1}{6.75}
\]

Thus:

\[
f''(2) \approx 0.148
\]

If you'd like, I can clarify the steps or provide more details.

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**Related Questions:**
1. How does the quotient rule work for general functions?
2. What is the process to simplify complex derivatives involving fractions?
3. Can we find \( f'''(x) \) from the function given here?
4. What are some real-world applications of second derivatives?
5. How do second derivatives inform us about concavity in graph analysis?

**Tip:** Always check if simplification before differentiating helps reduce calculation effort.

Learn how to find the second derivative of the function f(x) = x^2/(4 + x) at x = 2 using the quotient rule. This calculus problem walks through key steps of finding f'(x) and f''(x) and evaluates f''(2) approximately as 0.148. Ideal for advanced high school calculus students.

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