Derivative of (3x-2)/(3-x)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = 3 x - 2$ and $g{\left(x \right)} = 3 - x$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $3 x - 2$ term by term:

      1. The derivative of the constant $-2$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $3$

      The result is: $3$

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $3 - x$ term by term:

      1. The derivative of the constant $3$ is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

         1. Apply the power rule: $x$ goes to $1$

         So, the result is: $-1$

      The result is: $-1$

   Now plug in to the quotient rule:

   $\frac{7}{\left(3 - x\right)^{2}}$

2. Now simplify:

   $$\frac{7}{\left(x - 3\right)^{2}}$$

The answer is:

$$\frac{7}{\left(x - 3\right)^{2}}$$