Derivative of 2*x^6/3*x-8

1. Differentiate $x \frac{2 x^{6}}{3} - 8$ term by term:

   1. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = 2 x^{7}$ and $g{\left(x \right)} = 3$.

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

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

         1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$

         So, the result is: $14 x^{6}$

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

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

      Now plug in to the quotient rule:

      $\frac{14 x^{6}}{3}$

   2. The derivative of the constant $-8$ is zero.

   The result is: $\frac{14 x^{6}}{3}$

The answer is:

$$\frac{14 x^{6}}{3}$$