Derivative of 2+x*sqrt(3-x)

1. Differentiate $x \sqrt{3 - x} + 2$ term by term:

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

   2. 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 $1$

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

      1. Let $u = 3 - x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 - x\right)$:

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

            1. The derivative of the constant $3$ is zero.

            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: $-1$

            The result is: $-1$

         The result of the chain rule is:

         $$- \frac{1}{2 \sqrt{3 - x}}$$

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

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

2. Now simplify:

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

The answer is:

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