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

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

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

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

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

            1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$

            So, the result is: $28 x^{6}$

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

            1. Let $u = x^{3}$.

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

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

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

               The result of the chain rule is:

               $$- \frac{3}{x^{4}}$$

            So, the result is: $\frac{6}{x^{4}}$

         The result is: $28 x^{6} + \frac{6}{x^{4}}$

      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: $-5$

      The result is: $28 x^{6} - 5 + \frac{6}{x^{4}}$

   2. The derivative of the constant $2$ is zero.

   The result is: $28 x^{6} - 5 + \frac{6}{x^{4}}$

The answer is:

$$28 x^{6} - 5 + \frac{6}{x^{4}}$$