Use the Quotient Rule to find the derivative of the function.
f(x) = 
x
x − 6
 

To find the derivative of the function \( f(x) = \frac{x}{x - 6} \) using the Quotient Rule, we follow the formula for the derivative of a quotient of two functions:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}
\]

Here:
- \( u(x) = x \)
- \( v(x) = x - 6 \)

### Step 1: Differentiate \( u(x) \) and \( v(x) \)
- \( u'(x) = \frac{d}{dx}(x) = 1 \)
- \( v'(x) = \frac{d}{dx}(x - 6) = 1 \)

### Step 2: Apply the Quotient Rule
\[
f'(x) = \frac{(x - 6) \cdot 1 - x \cdot 1}{(x - 6)^2}
\]

Simplify the numerator:
\[
f'(x) = \frac{(x - 6) - x}{(x - 6)^2}
\]
\[
f'(x) = \frac{-6}{(x - 6)^2}
\]

Thus, the derivative of \( f(x) = \frac{x}{x - 6} \) is:
\[
f'(x) = \frac{-6}{(x - 6)^2}
\]

Let me know if you'd like more details on this solution!

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Here are 5 related questions to explore:
1. What happens to the derivative as \( x \) approaches 6?
2. Can the Quotient Rule be applied to any quotient of functions?
3. How would you find the second derivative of \( f(x) \)?
4. How is the Quotient Rule different from the Product Rule?
5. How would the chain rule apply if \( f(x) \) were a composition of functions?

**Tip:** The Quotient Rule is useful for rational functions, but sometimes rewriting the function and using the Product Rule can simplify the process!

Use the Quotient Rule to find the derivative of the function f(x) = x / (x − 6).

Learn how to find the derivative of the function f(x) = x / (x − 6) using the Quotient Rule. This step-by-step guide covers key calculus concepts and provides a simplified solution, suitable for students in Grades 11-12 or college-level calculus.

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