Derivative of y=sqrt(x^2-2x+5)

1. Let $u = x^{2} - 2 x + 5$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 2 x + 5\right)$:

   1. Differentiate $x^{2} - 2 x + 5$ term by term:

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

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

         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: $2$

         So, the result is: $-2$

      3. The derivative of the constant $5$ is zero.

      The result is: $2 x - 2$

   The result of the chain rule is:

   $$\frac{2 x - 2}{2 \sqrt{x^{2} - 2 x + 5}}$$

4. Now simplify:

   $$\frac{x - 1}{\sqrt{x^{2} - 2 x + 5}}$$

The answer is:

$$\frac{x - 1}{\sqrt{x^{2} - 2 x + 5}}$$