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The problem asks to find the second derivative of \( f(x) = 2 \sec^2(x) \).

### Step-by-Step Solution:

1. **First Derivative:**

   To find the second derivative, we first need to compute the first derivative of \( f(x) = 2 \sec^2(x) \). We'll use the chain rule and the known derivative of \( \sec^2(x) \).

   \[
   \frac{d}{dx}[\sec^2(x)] = 2 \sec^2(x) \tan(x)
   \]

   Therefore, the first derivative of \( f(x) \) is:

   \[
   f'(x) = 2 \cdot 2 \sec^2(x) \tan(x) = 4 \sec^2(x) \tan(x)
   \]

2. **Second Derivative:**

   Now, we need to differentiate \( f'(x) = 4 \sec^2(x) \tan(x) \). This requires the product rule since the expression is a product of \( \sec^2(x) \) and \( \tan(x) \).

   Let:
   \[
   u(x) = \sec^2(x) \quad \text{and} \quad v(x) = \tan(x)
   \]

   By the product rule:
   \[
   f''(x) = \frac{d}{dx}[u(x)v(x)] = u'(x) v(x) + u(x) v'(x)
   \]

   - \( u'(x) = 2 \sec^2(x) \tan(x) \)
   - \( v'(x) = \sec^2(x) \)

   Substituting these into the product rule:

   \[
   f''(x) = 4 \left( 2 \sec^2(x) \tan(x) \cdot \tan(x) + \sec^2(x) \cdot \sec^2(x) \right)
   \]

   Simplifying:

   \[
   f''(x) = 4 \left( 2 \sec^2(x) \tan^2(x) + \sec^4(x) \right)
   \]

   Factor out \( \sec^2(x) \):

   \[
   f''(x) = 4 \sec^2(x) \left( 2 \tan^2(x) + \sec^2(x) \right)
   \]

Thus, the second derivative is:

\[
f''(x) = 4 \sec^2(x) \left( 2 \tan^2(x) + \sec^2(x) \right)
\]

Would you like further details or have any questions on this?

### Related Questions:
1. How do you apply the chain rule to derivatives involving trigonometric functions?
2. What are the derivatives of basic trigonometric functions like \( \sec(x) \) and \( \tan(x) \)?
3. How do you apply the product rule in derivative calculations?
4. How do you simplify trigonometric expressions involving \( \sec(x) \) and \( \tan(x) \)?
5. What are the common applications of second derivatives in physics and engineering?

### Tip:
Always review the product and chain rules when working with complex trigonometric functions to avoid mistakes in multi-step differentiation.

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

Learn how to compute the second derivative of f(x) = 2 sec^2(x) using the product rule and chain rule. This detailed step-by-step solution walks you through the necessary formulas and techniques for higher-order derivatives involving trigonometric functions. Suitable for advanced calculus students.

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