Derivative of x^3-2x^6+6x

1. Differentiate $6 x + \left(- 2 x^{6} + x^{3}\right)$ term by term:

   1. Differentiate $- 2 x^{6} + x^{3}$ term by term:

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

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

         1. Apply the power rule: $x^{6}$ goes to $6 x^{5}$

         So, the result is: $- 12 x^{5}$

      The result is: $- 12 x^{5} + 3 x^{2}$

   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: $6$

   The result is: $- 12 x^{5} + 3 x^{2} + 6$

The answer is:

$$- 12 x^{5} + 3 x^{2} + 6$$