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: