Derivative of f(x)=x-2/x+3+5x

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

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

      1. Differentiate $x - \frac{2}{x}$ term by term:

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

         2. 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: $\frac{2}{x^{2}}$

         The result is: $1 + \frac{2}{x^{2}}$

      2. The derivative of the constant $3$ is zero.

      The result is: $1 + \frac{2}{x^{2}}$

   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: $6 + \frac{2}{x^{2}}$

The answer is:

$$6 + \frac{2}{x^{2}}$$