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We are tasked with finding the derivative of the function \( g(x) = 5 \sec(x) + \tan(x) \).

### Step-by-Step Derivative Calculation:

1. **Differentiate the first term \( 5 \sec(x) \):**
   - The derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \).
   - So, \( \frac{d}{dx} [5 \sec(x)] = 5 \sec(x) \tan(x) \).

2. **Differentiate the second term \( \tan(x) \):**
   - The derivative of \( \tan(x) \) is \( \sec^2(x) \).
   - So, \( \frac{d}{dx} [\tan(x)] = \sec^2(x) \).

### Putting it all together:
\[
g'(x) = 5 \sec(x) \tan(x) + \sec^2(x)
\]

This is the derivative of the given function \( g(x) \).

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

Here are some related questions you can explore:
1. What is the derivative of \( \sec(x) \) and why is it \( \sec(x) \tan(x) \)?
2. How do you differentiate products involving trigonometric functions?
3. Can you verify this result using the chain rule?
4. What is the geometric interpretation of the derivative of \( \tan(x) \)?
5. How does the derivative change if the function is \( g(x) = 5 \sec^2(x) + \tan(x) \)?

**Tip:** When working with trigonometric functions, it's useful to remember the key derivatives of \( \sin(x), \cos(x), \tan(x), \sec(x), \) and \( \csc(x) \).

Find the derivative of g when g(x) = 5sec(x) + tan(x).

Discover how to differentiate the function g(x) = 5sec(x) + tan(x) by applying trigonometric differentiation rules. This step-by-step solution explains the derivative of sec(x) and tan(x), suitable for students in college-level calculus.

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