Derivative of f(x)=4x^3-5x+3/x-1/2

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

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

      1. Differentiate $4 x^{3} - 5 x$ 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: $x^{3}$ goes to $3 x^{2}$

            So, the result is: $12 x^{2}$

         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: $12 x^{2} - 5$

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

      The result is: $12 x^{2} - 5 - \frac{3}{x^{2}}$

   2. The derivative of the constant $- \frac{1}{2}$ is zero.

   The result is: $12 x^{2} - 5 - \frac{3}{x^{2}}$

The answer is:

$$12 x^{2} - 5 - \frac{3}{x^{2}}$$