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

1. Apply the quotient rule, which is:

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

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

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

   1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

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

   1. Let $u = x - 2$.

   2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 2\right)$:

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

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

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

         The result is: $1$

      The result of the chain rule is:

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

   Now plug in to the quotient rule:

   $\frac{- 3 x^{3} \left(x - 2\right)^{2} + 3 x^{2} \left(x - 2\right)^{3}}{\left(x - 2\right)^{6}}$

2. Now simplify:

   $$- \frac{6 x^{2}}{\left(x - 2\right)^{4}}$$

The answer is:

$$- \frac{6 x^{2}}{\left(x - 2\right)^{4}}$$