Derivative of y=sqrt(x)*ln*x

1. Apply the product rule:

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

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

   1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

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

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

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

2. Now simplify:

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

The answer is:

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