Derivative of (x-3)sqrt(x)

1. Apply the product rule:

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

   $f{\left(x \right)} = x - 3$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $x - 3$ term by term:

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

      2. The derivative of the constant $-3$ is zero.

      The result is: $1$

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

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

   The result is: $\sqrt{x} + \frac{x - 3}{2 \sqrt{x}}$

2. Now simplify:

   $$\frac{3 \left(x - 1\right)}{2 \sqrt{x}}$$

The answer is:

$$\frac{3 \left(x - 1\right)}{2 \sqrt{x}}$$