Derivative of x*sqrt(9-x)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \sqrt{9 - x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 9 - x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(9 - x\right)$:

      1. Differentiate $9 - x$ term by term:

         1. The derivative of the constant $9$ 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:

      $$- \frac{1}{2 \sqrt{9 - x}}$$

   The result is: $- \frac{x}{2 \sqrt{9 - x}} + \sqrt{9 - x}$

2. Now simplify:

   $$\frac{3 \left(6 - x\right)}{2 \sqrt{9 - x}}$$

The answer is:

$$\frac{3 \left(6 - x\right)}{2 \sqrt{9 - x}}$$