Use derivative rules to compute the derivative of ln
(
sec(x) + tan(x)
)
. Simplify your
answer as much as you can

To compute the derivative of \( f(x) = \ln(\sec(x) + \tan(x)) \), we can apply the chain rule and logarithmic differentiation rules.

### Step-by-step Solution:

1. **Outer Function (Logarithm):**
   The derivative of \( \ln(u) \), where \( u \) is a function of \( x \), is:
   \[
   \frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx}
   \]
   So, here we apply this rule with \( u = \sec(x) + \tan(x) \).

   This gives:
   \[
   \frac{d}{dx} \ln(\sec(x) + \tan(x)) = \frac{1}{\sec(x) + \tan(x)} \cdot \frac{d}{dx} (\sec(x) + \tan(x))
   \]

2. **Derivative of the Inner Function (\( \sec(x) + \tan(x) \)):**
   The derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), and the derivative of \( \tan(x) \) is \( \sec^2(x) \). Therefore:
   \[
   \frac{d}{dx} (\sec(x) + \tan(x)) = \sec(x) \tan(x) + \sec^2(x)
   \]

3. **Putting it Together:**
   Now substitute this into the derivative from Step 1:
   \[
   \frac{d}{dx} \ln(\sec(x) + \tan(x)) = \frac{\sec(x) \tan(x) + \sec^2(x)}{\sec(x) + \tan(x)}
   \]

4. **Simplifying the Expression:**
   Notice that \( \sec^2(x) = \sec(x) \cdot \sec(x) \), so we can factor the numerator:
   \[
   \sec(x) \tan(x) + \sec^2(x) = \sec(x) (\tan(x) + \sec(x))
   \]

   So the derivative becomes:
   \[
   \frac{\sec(x) (\tan(x) + \sec(x))}{\sec(x) + \tan(x)}
   \]

   Since the numerator and denominator are both \( \sec(x) + \tan(x) \), they cancel out, leaving:
   \[
   \boxed{1}
   \]

### Final Answer:
The derivative of \( \ln(\sec(x) + \tan(x)) \) is \( 1 \).

Would you like to see more details or have any questions?

### Related Questions:
1. What are the chain rule and product rule in differentiation?
2. How do you differentiate \( \sec(x) \) and \( \tan(x) \)?
3. How would you differentiate a more complicated logarithmic function like \( \ln(f(x)) \)?
4. What are some practical applications of derivatives in real life?
5. How do you apply logarithmic differentiation to more complex functions?

### Tip:
Always remember to simplify the expression as much as possible after applying derivative rules; sometimes terms cancel out, leading to a simpler result.

Use derivative rules to compute the derivative of ln(sec(x) + tan(x)). Simplify your answer as much as you can.

Learn how to compute the derivative of ln(sec(x) + tan(x)) using the chain rule and logarithmic differentiation. The solution covers key differentiation techniques, including derivatives of sec(x) and tan(x). Suitable for students in Grades 11-12 and college calculus courses.

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