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We are given the function \( f(x) = \frac{1}{x} \cdot \cos x \) and need to find the derivative \( f'(x) \).

### Step-by-step solution:
This is a product of two functions:
\[
f(x) = \frac{1}{x} \cdot \cos x
\]
We will use the product rule for derivatives, which states:
\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]
Here, \( f(x) = \frac{1}{x} \) and \( g(x) = \cos x \).

1. Derivative of \( \frac{1}{x} \):
\[
\frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}
\]

2. Derivative of \( \cos x \):
\[
\frac{d}{dx} \left( \cos x \right) = -\sin x
\]

Now, applying the product rule:
\[
f'(x) = \left( -\frac{1}{x^2} \right) \cdot \cos x + \frac{1}{x} \cdot \left( -\sin x \right)
\]
Simplify the expression:
\[
f'(x) = -\frac{\cos x}{x^2} - \frac{\sin x}{x}
\]

To combine these terms under a common denominator:
\[
f'(x) = \frac{-\cos x - x \sin x}{x^2}
\]

Thus, the derivative is:
\[
f'(x) = \frac{-\cos x - x \sin x}{x^2}
\]

### Correct answer:
This matches option **C**:
\[
\frac{-\cos x - x \sin x}{x^2}
\]

Would you like more details or have any other questions? 

Here are five related questions to consider:
1. How do you apply the quotient rule for derivatives?
2. What are some common mistakes when using the product rule?
3. Can you use the chain rule in combination with the product rule?
4. How does the derivative of trigonometric functions behave?
5. What is the geometric interpretation of the derivative of this function?

**Tip**: Always simplify expressions after applying derivative rules to avoid unnecessary complexity in final answers.

If f(x) = \( \frac{1}{x} \cdot \cos x \), then f'(x) = ?

Learn how to find the derivative of the function f(x) = 1/x * cos(x) using the product rule. This problem involves key calculus concepts such as the product rule, derivatives of trigonometric functions, and simplification techniques. Ideal for students in Grades 11-12.

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