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We are given \( f(x) = \sec x \) and asked to find the second derivative \( f''(x) \).

### Step 1: Find the first derivative \( f'(x) \)
The derivative of \( \sec x \) is:
\[
f'(x) = \frac{d}{dx}(\sec x) = \sec x \tan x
\]

### Step 2: Find the second derivative \( f''(x) \)
Now, differentiate \( f'(x) = \sec x \tan x \) using the product rule:
\[
f''(x) = \frac{d}{dx}(\sec x \tan x)
\]
Applying the product rule:
\[
f''(x) = \frac{d}{dx}(\sec x) \cdot \tan x + \sec x \cdot \frac{d}{dx}(\tan x)
\]

From earlier, we know:
\[
\frac{d}{dx}(\sec x) = \sec x \tan x
\]
and the derivative of \( \tan x \) is \( \sec^2 x \).

Thus:
\[
f''(x) = (\sec x \tan x) \cdot \tan x + \sec x \cdot \sec^2 x
\]

Simplifying:
\[
f''(x) = \sec x \tan^2 x + \sec^3 x
\]

So, the second derivative is:
\[
f''(x) = \sec x (\tan^2 x + \sec^2 x)
\]

### Final Answer:
\[
f''(x) = \sec x (\sec^2 x)
\]
\[
f''(x) = \sec^3 x
\]

Would you like any further details or explanations?

Here are 5 related questions for deeper understanding:
1. What is the third derivative of \( f(x) = \sec x \)?
2. How can you apply the chain rule to find derivatives of trigonometric functions?
3. What is the integral of \( \sec x \)?
4. How can we find higher-order derivatives for other trigonometric functions like \( \sin x \) or \( \cos x \)?
5. Can you explain the product rule and how it was used in this problem?

**Tip**: Always remember to apply the product rule when differentiating the product of two functions.

Learn how to find the second derivative of f(x) = sec(x) using differentiation techniques. This example explores the product rule and trigonometric derivatives, perfect for students in Grades 11-12.

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