Derivative of 3ln2x-1/x+2

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

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

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

         1. Let $u = 2 x$.

         2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

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

            1. 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: $2$

            The result of the chain rule is:

            $$\frac{1}{x}$$

         So, the result is: $\frac{3}{x}$

      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: $\frac{1}{x^{2}}$

      The result is: $\frac{3}{x} + \frac{1}{x^{2}}$

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

   The result is: $\frac{3}{x} + \frac{1}{x^{2}}$

2. Now simplify:

   $$\frac{3 x + 1}{x^{2}}$$

The answer is:

$$\frac{3 x + 1}{x^{2}}$$