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We are tasked with finding the derivative of the function \(\frac{\cos^2(x)}{x}\). To solve this, we can apply both the **quotient rule** and the **chain rule** of differentiation. Here's a step-by-step breakdown:

### Step 1: Apply the Quotient Rule
The quotient rule states that for a function \( f(x) = \frac{u(x)}{v(x)} \), the derivative is:
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{v(x)^2}
\]
In our case:
- \( u(x) = \cos^2(x) \)
- \( v(x) = x \)

### Step 2: Differentiate \( u(x) = \cos^2(x) \)
To differentiate \( u(x) \), we need to apply the chain rule:
\[
u'(x) = 2\cos(x) \cdot (-\sin(x)) = -2\cos(x)\sin(x)
\]

### Step 3: Differentiate \( v(x) = x \)
The derivative of \( v(x) \) is straightforward:
\[
v'(x) = 1
\]

### Step 4: Apply the Quotient Rule
Now we apply the quotient rule formula:
\[
\frac{d}{dx} \left( \frac{\cos^2(x)}{x} \right) = \frac{x \cdot (-2\cos(x)\sin(x)) - \cos^2(x) \cdot 1}{x^2}
\]
Simplifying the numerator:
\[
= \frac{-2x\cos(x)\sin(x) - \cos^2(x)}{x^2}
\]

### Step 5: Factor and Simplify
We can factor out \(-\cos(x)\) from the terms in the numerator:
\[
= \frac{-\cos(x)(2x\sin(x) + \cos(x))}{x^2}
\]

### Final Answer:
This matches with option **C** from the multiple-choice answers:
\[
\boxed{C}
\]

Would you like any further details or clarification?

### Related Questions:
1. How does the quotient rule apply to different rational functions?
2. How do you use the chain rule when dealing with powers of trigonometric functions?
3. What are the steps for finding critical points of functions like \(\frac{\cos^2(x)}{x}\)?
4. How can you determine if the derivative of a function is positive or negative on specific intervals?
5. How do you evaluate limits of functions involving trigonometric expressions?

**Tip:** When applying the quotient rule, always make sure to simplify both the numerator and the denominator to avoid algebraic mistakes.

Learn how to find the derivative of cos^2(x) / x step by step using the Quotient Rule and Chain Rule. This problem involves applying the chain rule to trigonometric functions and differentiating rational expressions. Suitable for high school calculus students.

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