Derivative of 10/x^(2/3)+15*x

1. Differentiate $15 x + \frac{10}{x^{\frac{2}{3}}}$ term by term:

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

      1. Let $u = x^{\frac{2}{3}}$.

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

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

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

         The result of the chain rule is:

         $$- \frac{2}{3 x^{\frac{5}{3}}}$$

      So, the result is: $- \frac{20}{3 x^{\frac{5}{3}}}$

   2. 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: $15$

   The result is: $15 - \frac{20}{3 x^{\frac{5}{3}}}$

The answer is:

$$15 - \frac{20}{3 x^{\frac{5}{3}}}$$