Derivative of y=x*ln(x/2)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

   $g{\left(x \right)} = \log{\left(\frac{x}{2} \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \frac{x}{2}$.

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

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

      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: $\frac{1}{2}$

      The result of the chain rule is:

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

   The result is: $\log{\left(\frac{x}{2} \right)} + 1$

2. Now simplify:

   $$\log{\left(\frac{x}{2} \right)} + 1$$

The answer is:

$$\log{\left(\frac{x}{2} \right)} + 1$$