Derivative of 2*log(x)-x/2+1

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

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

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

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

         So, the result is: $\frac{2}{x}$

      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: $- \frac{1}{2}$

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

   2. The derivative of the constant $1$ is zero.

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

2. Now simplify:

   $$\frac{4 - x}{2 x}$$

The answer is: