Derivative of g(x)=8-4x^3+2x^8

1. Differentiate $2 x^{8} - 4 x^{3} + 8$ term by term:

   1. The derivative of the constant $8$ is zero.

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

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

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

         So, the result is: $12 x^{2}$

      So, the result is: $- 12 x^{2}$

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

      1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$

      So, the result is: $16 x^{7}$

   The result is: $16 x^{7} - 12 x^{2}$

2. Now simplify:

   $$x^{2} \cdot \left(16 x^{5} - 12\right)$$

The answer is:

$$x^{2} \cdot \left(16 x^{5} - 12\right)$$