Derivative of y=x+√x/x-2x^1/3

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

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

      1. Apply the power rule: $x$ goes to $1$

      2. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = \sqrt{x}$ and $g{\left(x \right)} = x$.

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

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

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

         1. Apply the power rule: $x$ goes to $1$

         Now plug in to the quotient rule:

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

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

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

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

      So, the result is: $- \frac{2}{3 x^{\frac{2}{3}}}$

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

The answer is:

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