Derivative of x^5-5/x^4+(5x)^(1/3)

1. Differentiate $x^{5} + \sqrt[3]{5 x} - \frac{5}{x^{4}}$ term by term:

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

   2. 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 $u = x^{4}$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{4}$:

            1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

            The result of the chain rule is:

            $$- \frac{4}{x^{5}}$$

         So, the result is: $- \frac{20}{x^{5}}$

      So, the result is: $\frac{20}{x^{5}}$

   3. Let $u = 5 x$.

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

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

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

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

         So, the result is: $5$

      The result of the chain rule is:

      $$\frac{\sqrt[3]{5}}{3 x^{\frac{2}{3}}}$$

   The result is: $5 x^{4} + \frac{20}{x^{5}} + \frac{\sqrt[3]{5}}{3 x^{\frac{2}{3}}}$

The answer is:

$$5 x^{4} + \frac{20}{x^{5}} + \frac{\sqrt[3]{5}}{3 x^{\frac{2}{3}}}$$