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!