Derivative of (x-1)√x^2-2x+2

1. Differentiate $\left(\left(x - 1\right) \left(\sqrt{x}\right)^{2} - 2 x\right) + 2$ term by term:

   1. Differentiate $\left(x - 1\right) \left(\sqrt{x}\right)^{2} - 2 x$ term by term:

      1. Apply the product rule:

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

         $f{\left(x \right)} = x - 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

            The result is: $1$

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

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

         2. Apply the power rule: $u^{2}$ goes to $2 u$

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$:

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

            The result of the chain rule is:

            $$1$$

         The result is: $\left(\sqrt{x}\right)^{2} + x - 1$

      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: $-2$

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

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

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

2. Now simplify:

   $$2 x - 3$$

The answer is:

$$2 x - 3$$