Derivative of y=sqrt(x)^3+2/x-4/x^5-5x^3

1. Differentiate $\left(\sqrt{x}\right)^{3} - 5 x^{3} - \frac{4}{x^{5}} + \frac{2}{x}$ term by term:

   1. Let $u = \sqrt{x}$.

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

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

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

      The result of the chain rule is:

      $$\frac{3 \sqrt{x}}{2}$$

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

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

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

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

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

         1. Let $u = x^{5}$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{5}$:

            1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

            The result of the chain rule is:

            $$- \frac{5}{x^{6}}$$

         So, the result is: $- \frac{20}{x^{6}}$

      So, the result is: $\frac{20}{x^{6}}$

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

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

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

         So, the result is: $15 x^{2}$

      So, the result is: $- 15 x^{2}$

   The result is: $\frac{3 \sqrt{x}}{2} - 15 x^{2} - \frac{2}{x^{2}} + \frac{20}{x^{6}}$

2. Now simplify:

   $$\frac{3 x^{\frac{13}{2}} - 30 x^{8} - 4 x^{4} + 40}{2 x^{6}}$$

The answer is:

$$\frac{3 x^{\frac{13}{2}} - 30 x^{8} - 4 x^{4} + 40}{2 x^{6}}$$