To find the derivative of the function \( f(x) = x \tan(x) \sec(x) \), we'll use the product rule and the chain rule.

### Step 1: Identify the components
We have \( f(x) = x \cdot \tan(x) \cdot \sec(x) \).

### Step 2: Apply the Product Rule
The product rule states that if \( f(x) = u(x) \cdot v(x) \cdot w(x) \), then:
\[
f'(x) = u'(x) \cdot v(x) \cdot w(x) + u(x) \cdot v'(x) \cdot w(x) + u(x) \cdot v(x) \cdot w'(x)
\]
Here, \( u(x) = x \), \( v(x) = \tan(x) \), and \( w(x) = \sec(x) \).

### Step 3: Find the derivatives of each component
1. \( u'(x) = 1 \)
2. \( v'(x) = \sec^2(x) \) (since the derivative of \( \tan(x) \) is \( \sec^2(x) \))
3. \( w'(x) = \sec(x) \tan(x) \) (since the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \))

### Step 4: Apply the product rule formula
Now, substituting the derivatives into the formula:
\[
f'(x) = 1 \cdot \tan(x) \cdot \sec(x) + x \cdot \sec^2(x) \cdot \sec(x) + x \cdot \tan(x) \cdot \sec(x) \cdot \tan(x)
\]

Simplify each term:
1. \( \tan(x) \cdot \sec(x) \)
2. \( x \cdot \sec^3(x) \)
3. \( x \cdot \sec(x) \cdot \tan^2(x) \)

Thus, the derivative \( f'(x) \) is:
\[
f'(x) = \tan(x) \sec(x) + x \sec^3(x) + x \sec(x) \tan^2(x)
\]

This is the derivative of \( f(x) \).

Do you want more details or have any questions?

Here are 5 related questions to consider:
1. How do you apply the product rule in other complex functions?
2. What is the significance of the chain rule in differentiation?
3. How would you find the second derivative \( f''(x) \) of this function?
4. Can you use trigonometric identities to simplify the derivative further?
5. What happens if you evaluate \( f'(x) \) at specific values of \( x \)?

**Tip:** When differentiating products of multiple functions, always apply the product rule step by step, ensuring you handle each part carefully.

Learn how to differentiate the function f(x) = x tan(x) sec(x) using the product rule and trigonometric identities. This problem involves advanced differentiation techniques and is suitable for advanced high school and college-level students.

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