Derivative of 8x-16/(x-2)^3

1. Differentiate $8 x - \frac{16}{\left(x - 2\right)^{3}}$ term by term:

   1. 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: $8$

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

      1. Let $u = \left(x - 2\right)^{3}$.

      2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

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

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

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

               The result is: $1$

            The result of the chain rule is:

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

         The result of the chain rule is:

         $$- \frac{3}{\left(x - 2\right)^{4}}$$

      So, the result is: $\frac{48}{\left(x - 2\right)^{4}}$

   The result is: $8 + \frac{48}{\left(x - 2\right)^{4}}$

2. Now simplify:

   $$8 + \frac{48}{\left(x - 2\right)^{4}}$$

The answer is:

$$8 + \frac{48}{\left(x - 2\right)^{4}}$$