Derivative of (sqrt(x-2sqrt(x-2)))

1. Let $u = x - 2 \sqrt{x - 2}$.

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

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

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

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

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

         1. Let $u = x - 2$.

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

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

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

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

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

               The result is: $1$

            The result of the chain rule is:

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

         So, the result is: $- \frac{1}{\sqrt{x - 2}}$

      The result is: $1 - \frac{1}{\sqrt{x - 2}}$

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is:

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