Derivative of x^(3/5)-8/x^4+2x^(5/7)

1. Differentiate $2 x^{\frac{5}{7}} + x^{\frac{3}{5}} - \frac{8}{x^{4}}$ term by term:

   1. Apply the power rule: $x^{\frac{3}{5}}$ goes to $\frac{3}{5 x^{\frac{2}{5}}}$

   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: $- \frac{32}{x^{5}}$

      So, the result is: $\frac{32}{x^{5}}$

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

      1. Apply the power rule: $x^{\frac{5}{7}}$ goes to $\frac{5}{7 x^{\frac{2}{7}}}$

      So, the result is: $\frac{10}{7 x^{\frac{2}{7}}}$

   The result is: $\frac{32}{x^{5}} + \frac{3}{5 x^{\frac{2}{5}}} + \frac{10}{7 x^{\frac{2}{7}}}$

The answer is:

$$\frac{32}{x^{5}} + \frac{3}{5 x^{\frac{2}{5}}} + \frac{10}{7 x^{\frac{2}{7}}}$$