Use the Quotient Rule to find the derivative of the function.
f(x) = x/x − 4
 

To differentiate the function \( f(x) = \frac{x}{x - 4} \), we can apply the **Quotient Rule**.

The Quotient Rule is given by:

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

Here, we identify the following components:
- \( u(x) = x \)
- \( v(x) = x - 4 \)

### Step 1: Differentiate \( u(x) \) and \( v(x) \)
- \( u'(x) = 1 \)
- \( v'(x) = 1 \)

### Step 2: Apply the Quotient Rule

Using the Quotient Rule formula:

\[
f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}
\]

Substitute the values:

\[
f'(x) = \frac{(1)(x - 4) - (x)(1)}{(x - 4)^2}
\]

Simplify the numerator:

\[
f'(x) = \frac{x - 4 - x}{(x - 4)^2} = \frac{-4}{(x - 4)^2}
\]

### Final Answer:

\[
f'(x) = \frac{-4}{(x - 4)^2}
\]

This is the derivative of the function \( f(x) = \frac{x}{x - 4} \).

Would you like more details or have any further questions?

Here are some related questions to deepen your understanding:
1. How does the Quotient Rule differ from the Product Rule in differentiation?
2. Can you apply the Quotient Rule to more complex functions?
3. What happens to the derivative as \( x \) approaches 4 in this function?
4. Can the Chain Rule be combined with the Quotient Rule in some cases?
5. How would you differentiate a function with a trigonometric component using the Quotient Rule?

**Tip:** Always simplify the function as much as possible before applying differentiation rules, to minimize errors!

Use the Quotient Rule to find the derivative of the function. f(x) = x / (x - 4)

Learn how to apply the Quotient Rule to find the derivative of the function f(x) = x / (x - 4). This step-by-step solution covers key calculus concepts and the use of differentiation formulas. Suitable for students in Grades 11-12 or early college.

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