To find the second derivative of \( f(x) = \frac{1}{(x^2 + 4)^2} \), let's go step by step.

### Step 1: First Derivative

We can use the chain rule and quotient rule to find the first derivative.

We rewrite \( f(x) = (x^2 + 4)^{-2} \).

Now, differentiate using the chain rule:

\[
f'(x) = -2(x^2 + 4)^{-3} \cdot (2x)
\]

Simplifying:

\[
f'(x) = \frac{-4x}{(x^2 + 4)^3}
\]

### Step 2: Second Derivative

Now, to find the second derivative \( f''(x) \), we differentiate \( f'(x) \).

Differentiate \( f'(x) = \frac{-4x}{(x^2 + 4)^3} \) using the quotient rule.

Let:
- \( u = -4x \)
- \( v = (x^2 + 4)^3 \)

Using the quotient rule \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), we need to find \( u' \) and \( v' \).

- \( u' = -4 \)
- \( v' = 3(x^2 + 4)^2 \cdot (2x) = 6x(x^2 + 4)^2 \)

Now, applying the quotient rule:

\[
f''(x) = \frac{(-4)(x^2 + 4)^3 - (-4x)(6x(x^2 + 4)^2)}{(x^2 + 4)^6}
\]

Simplifying:

\[
f''(x) = \frac{-4(x^2 + 4)^3 + 24x^2(x^2 + 4)^2}{(x^2 + 4)^6}
\]

Factor out \( (x^2 + 4)^2 \) from the numerator:

\[
f''(x) = \frac{(x^2 + 4)^2 \left[ -4(x^2 + 4) + 24x^2 \right]}{(x^2 + 4)^6}
\]

Simplify the expression inside the brackets:

\[
-4(x^2 + 4) + 24x^2 = -4x^2 - 16 + 24x^2 = 20x^2 - 16
\]

Now the second derivative becomes:

\[
f''(x) = \frac{(x^2 + 4)^2 (20x^2 - 16)}{(x^2 + 4)^6}
\]

Canceling out \( (x^2 + 4)^2 \):

\[
f''(x) = \frac{20x^2 - 16}{(x^2 + 4)^4}
\]

Thus, the second derivative is:

\[
f''(x) = \frac{20x^2 - 16}{(x^2 + 4)^4}
\]

Would you like more details on any step or clarification?

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Here are 5 related questions:
1. What is the third derivative of \( f(x) = \frac{1}{(x^2 + 4)^2} \)?
2. How would you compute higher-order derivatives using general rules?
3. Can you find the critical points of \( f(x) = \frac{1}{(x^2 + 4)^2} \)?
4. How do second derivatives help identify concavity in functions?
5. What does the second derivative test indicate about local extrema?

**Tip:** The quotient rule can sometimes be avoided by simplifying the function into a product form, using the chain rule directly.

Discover how to compute the second derivative of f(x) = 1 / (x^2 + 4)^2 using the chain rule and quotient rule. This step-by-step solution walks through key calculus concepts including the first and second derivatives. Ideal for college calculus students.

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