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The given function is \( f(x) = 2 \cot^2(x) \). We are asked to find the second derivative, \( f''(x) \).

### Step 1: First Derivative
Start by finding the first derivative of \( f(x) = 2 \cot^2(x) \) using the chain rule.

1. Use the chain rule for \( \cot^2(x) \). Let \( u(x) = \cot(x) \), so \( f(x) = 2 u(x)^2 \).
2. The derivative of \( 2 u(x)^2 \) is \( 4 u(x) u'(x) \), where \( u'(x) = -\csc^2(x) \) (the derivative of \( \cot(x) \)).
3. Substituting back, we get:

\[
f'(x) = 4 \cot(x) (-\csc^2(x)) = -4 \cot(x) \csc^2(x)
\]

### Step 2: Second Derivative
Now, find the derivative of \( f'(x) = -4 \cot(x) \csc^2(x) \).

We need to apply the product rule here:
- First function: \( -4 \cot(x) \)
- Second function: \( \csc^2(x) \)

The derivative of \( \cot(x) \) is \( -\csc^2(x) \), and the derivative of \( \csc^2(x) \) is \( -2 \csc^2(x) \cot(x) \) (by the chain rule).

Using the product rule:

\[
f''(x) = -4 \left[ \csc^2(x) \cdot (-\csc^2(x)) + \cot(x) \cdot (-2 \csc^2(x) \cot(x)) \right]
\]

Simplifying each term:

1. \( \csc^2(x) \cdot (-\csc^2(x)) = -\csc^4(x) \)
2. \( \cot(x) \cdot (-2 \csc^2(x) \cot(x)) = -2 \csc^2(x) \cot^2(x) \)

Thus, we get:

\[
f''(x) = -4 \left[ -\csc^4(x) - 2 \csc^2(x) \cot^2(x) \right]
\]

Simplifying further:

\[
f''(x) = 4 \csc^4(x) + 8 \csc^2(x) \cot^2(x)
\]

### Conclusion:
The second derivative is \( f''(x) = 4 \csc^4(x) + 8 \csc^2(x) \cot^2(x) \), which matches the third option in the list.

Would you like more details, or have any questions on this? Here are some related questions to expand on this topic:

1. How does the chain rule apply to trigonometric functions?
2. What is the product rule and when do we use it in calculus?
3. How can you simplify higher-order trigonometric derivatives?
4. What are some common mistakes to avoid when differentiating trigonometric functions?
5. Can the second derivative be used to determine concavity for trigonometric functions?

**Tip:** Always write down intermediate steps clearly when applying the product and chain rules to avoid sign or simplification errors.

Find the second derivative of f(x) = 2 cot^2(x).

Find the second derivative of the trigonometric function f(x) = 2 cot^2(x) using the chain and product rules. Detailed step-by-step explanation for AP Calculus or college-level calculus students.

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