We are asked to find the derivative of the expression:

\[
f(x) = \frac{4}{\cos(x)} + \frac{1}{\tan(x)}
\]

### Step 1: Rewrite the expression
We can rewrite the terms using known trigonometric identities:
\[
f(x) = 4 \sec(x) + \cot(x)
\]
where \( \sec(x) = \frac{1}{\cos(x)} \) and \( \cot(x) = \frac{1}{\tan(x)} \).

### Step 2: Derivatives of the terms
- The derivative of \( 4\sec(x) \) is \( 4\sec(x)\tan(x) \) because \( \frac{d}{dx} [\sec(x)] = \sec(x)\tan(x) \).
- The derivative of \( \cot(x) \) is \( -\csc^2(x) \) because \( \frac{d}{dx} [\cot(x)] = -\csc^2(x) \).

### Step 3: Apply the derivatives
Now, we differentiate term by term:
\[
\frac{d}{dx} f(x) = 4 \sec(x)\tan(x) - \csc^2(x)
\]

### Final result:
The derivative of \( f(x) \) is:
\[
f'(x) = 4 \sec(x) \tan(x) - \csc^2(x)
\]

Let me know if you need further details or clarifications!

Here are 5 related questions:
1. What are the basic trigonometric identities used in calculus?
2. How do you find the derivative of a product of functions?
3. How do you differentiate \( \sec(x) \) and \( \csc(x) \)?
4. What is the chain rule, and how does it apply to trigonometric functions?
5. How would you compute the second derivative of this function?

**Tip:** When differentiating trigonometric functions, remember to use known derivatives like \( \frac{d}{dx} [\sin(x)] = \cos(x) \) and their reciprocals for secant, cosecant, and cotangent functions.

This page provides a detailed solution for finding the derivative of 4/cos(x) + 1/tan(x), utilizing trigonometric identities and derivative rules. Key concepts include the differentiation of secant and cotangent functions, making it suitable for high school calculus and college-level math students.

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