Derivative of cbrt(x)+1/x-3/x^2

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

   1. Differentiate $\sqrt[3]{x} + \frac{1}{x}$ term by term:

      1. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$

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

      The result is: $- \frac{1}{x^{2}} + \frac{1}{3 x^{\frac{2}{3}}}$

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

      1. Let $u = x^{2}$.

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

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

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

         The result of the chain rule is:

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

      So, the result is: $\frac{6}{x^{3}}$

   The result is: $- \frac{1}{x^{2}} + \frac{6}{x^{3}} + \frac{1}{3 x^{\frac{2}{3}}}$

The answer is:

$$- \frac{1}{x^{2}} + \frac{6}{x^{3}} + \frac{1}{3 x^{\frac{2}{3}}}$$