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To differentiate the given function \( y = \frac{6 - \sec(x)}{\tan(x)} \), we will apply the **quotient rule** for differentiation.

### Quotient Rule:
If \( y = \frac{u(x)}{v(x)} \), then  
\[
\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}.
\]
Here:  
\( u(x) = 6 - \sec(x) \) and \( v(x) = \tan(x) \).

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### Step 1: Differentiate \( u(x) = 6 - \sec(x) \)
\[
u'(x) = 0 - \sec(x) \tan(x) = -\sec(x) \tan(x).
\]

---

### Step 2: Differentiate \( v(x) = \tan(x) \)
\[
v'(x) = \sec^2(x).
\]

---

### Step 3: Apply the Quotient Rule
\[
\frac{dy}{dx} = \frac{\tan(x) \cdot \left(-\sec(x) \tan(x)\right) - (6 - \sec(x)) \cdot \sec^2(x)}{\tan^2(x)}.
\]

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### Step 4: Simplify the Numerator
\[
\text{Numerator} = -\sec(x) \tan^2(x) - 6 \sec^2(x) + \sec^3(x).
\]
So,  
\[
\frac{dy}{dx} = \frac{-\sec(x) \tan^2(x) - 6 \sec^2(x) + \sec^3(x)}{\tan^2(x)}.
\]

This is the derivative, and it can be further simplified depending on the desired form, but this expression is a correct answer.

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Do you want me to further simplify or verify anything specific? 

Here are five related questions you might find useful:
1. How do we simplify the final derivative more efficiently?
2. What are some common applications of the quotient rule in trigonometric functions?
3. How do trigonometric identities help in simplifying derivatives?
4. Can this function be rewritten in a form that makes differentiation easier?
5. What are the limits of the function \( y = \frac{6 - \sec(x)}{\tan(x)} \) as \( x \to 0 \) or \( x \to \pi/2 \)?

**Tip:** Always check the domain of trigonometric functions to ensure the derivative is valid over the given interval.

Differentiate the function y = (6 - sec(x)) / tan(x).

Explore a step-by-step differentiation of the function y = (6 - sec(x)) / tan(x). This problem applies the quotient rule along with derivatives of trigonometric functions, such as sec(x) and tan(x). Perfect for students in Grades 11-12 or early college calculus courses.

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