Derivative of -((1)/(3))(3-x)sqrt(2x-3)

1. Apply the product rule:

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

   $f{\left(x \right)} = - 0.333333333333333 \left(3 - x\right)$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. The derivative of a constant times a function is the constant times the derivative of the function.

      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: $-1$

      So, the result is: $0.333333333333333$

   $g{\left(x \right)} = \sqrt{2 x - 3}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 2 x - 3$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 3\right)$:

      1. Differentiate $2 x - 3$ term by term:

         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: $2$

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

         The result is: $2$

      The result of the chain rule is:

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

   The result is: $- \frac{0.333333333333333 \left(3 - x\right)}{\sqrt{2 x - 3}} + 0.333333333333333 \sqrt{2 x - 3}$

2. Now simplify:

   $$\frac{1.0 x - 2.0}{\sqrt{2 x - 3}}$$

The answer is:

$$\frac{1.0 x - 2.0}{\sqrt{2 x - 3}}$$