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Let's go through the problem step by step. The instruction says **"Tentukan Turunan Fungsi Berikut"** which means "Determine the derivative of the following functions."

Here are the functions for which derivatives need to be found:

1. \( f(x) = \frac{x + 2}{x - 4} \)
2. \( f(x) = \frac{-4x}{5x + 1} \)
3. \( f(x) = \frac{x^2 - x + 1}{x - 3} \)

### Solution using the Quotient Rule:
For each of these functions, we will apply the **quotient rule** of differentiation, which is given by:
\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) u'(x) - u(x) v'(x)}{[v(x)]^2}
\]
Where:
- \( u(x) \) is the numerator function.
- \( v(x) \) is the denominator function.
- \( u'(x) \) and \( v'(x) \) are their respective derivatives.

### 1. \( f(x) = \frac{x + 2}{x - 4} \)
- \( u(x) = x + 2 \), so \( u'(x) = 1 \)
- \( v(x) = x - 4 \), so \( v'(x) = 1 \)

Now applying the quotient rule:
\[
f'(x) = \frac{(x - 4)(1) - (x + 2)(1)}{(x - 4)^2}
\]
Simplifying:
\[
f'(x) = \frac{x - 4 - x - 2}{(x - 4)^2} = \frac{-6}{(x - 4)^2}
\]
So, \( f'(x) = \frac{-6}{(x - 4)^2} \).

### 2. \( f(x) = \frac{-4x}{5x + 1} \)
- \( u(x) = -4x \), so \( u'(x) = -4 \)
- \( v(x) = 5x + 1 \), so \( v'(x) = 5 \)

Applying the quotient rule:
\[
f'(x) = \frac{(5x + 1)(-4) - (-4x)(5)}{(5x + 1)^2}
\]
Simplifying:
\[
f'(x) = \frac{-20x - 4 + 20x}{(5x + 1)^2} = \frac{-4}{(5x + 1)^2}
\]
So, \( f'(x) = \frac{-4}{(5x + 1)^2} \).

### 3. \( f(x) = \frac{x^2 - x + 1}{x - 3} \)
- \( u(x) = x^2 - x + 1 \), so \( u'(x) = 2x - 1 \)
- \( v(x) = x - 3 \), so \( v'(x) = 1 \)

Applying the quotient rule:
\[
f'(x) = \frac{(x - 3)(2x - 1) - (x^2 - x + 1)(1)}{(x - 3)^2}
\]
Expanding:
\[
f'(x) = \frac{(2x^2 - x - 6x + 3) - (x^2 - x + 1)}{(x - 3)^2}
= \frac{2x^2 - 7x + 3 - x^2 + x - 1}{(x - 3)^2}
\]
Simplifying:
\[
f'(x) = \frac{x^2 - 6x + 2}{(x - 3)^2}
\]
So, \( f'(x) = \frac{x^2 - 6x + 2}{(x - 3)^2} \).

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Do you want further clarification on any part of this solution or additional details? 

Here are 5 related questions:
1. How does the quotient rule apply when the denominator is a constant?
2. What are the conditions under which a derivative does not exist?
3. How do you simplify expressions after applying the quotient rule?
4. What is the relationship between derivatives and slopes of tangent lines?
5. How can you use the chain rule along with the quotient rule?

**Tip**: Always simplify the derivative expression as much as possible after applying the quotient rule to avoid mistakes in further calculations.

Tentukan turunan fungsi berikut:
1. f(x) = (x + 2) / (x - 4)
2. f(x) = (-4x) / (5x + 1)
3. f(x) = (x^2 - x + 1) / (x - 3)

Learn how to find the derivatives of rational functions such as (x + 2) / (x - 4), (-4x) / (5x + 1), and (x^2 - x + 1) / (x - 3) using the quotient rule. This problem is ideal for Grade 11-12 students learning calculus and differentiation.

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