Derivative of (2/x^3)-(x^6/9)+12sqrt(x)

1. Differentiate $12 \sqrt{x} + \left(- \frac{x^{6}}{9} + \frac{2}{x^{3}}\right)$ term by term:

   1. Differentiate $- \frac{x^{6}}{9} + \frac{2}{x^{3}}$ term by term:

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

         1. Let $u = x^{3}$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{3}$:

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

            The result of the chain rule is:

            $$- \frac{3}{x^{4}}$$

         So, the result is: $- \frac{6}{x^{4}}$

      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: $- \frac{2 x^{5}}{3}$

      The result is: $- \frac{2 x^{5}}{3} - \frac{6}{x^{4}}$

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

      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

      So, the result is: $\frac{6}{\sqrt{x}}$

   The result is: $- \frac{2 x^{5}}{3} - \frac{6}{x^{4}} + \frac{6}{\sqrt{x}}$

The answer is:

$$- \frac{2 x^{5}}{3} - \frac{6}{x^{4}} + \frac{6}{\sqrt{x}}$$