Derivative of sqrt(x^2-6x+13)

1. Let $u = \left(x^{2} - 6 x\right) + 13$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} - 6 x\right) + 13\right)$:

   1. Differentiate $\left(x^{2} - 6 x\right) + 13$ term by term:

      1. Differentiate $x^{2} - 6 x$ 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. Apply the power rule: $x$ goes to $1$

            So, the result is: $-6$

         The result is: $2 x - 6$

      2. The derivative of the constant $13$ is zero.

      The result is: $2 x - 6$

   The result of the chain rule is:

   $$\frac{2 x - 6}{2 \sqrt{\left(x^{2} - 6 x\right) + 13}}$$

4. Now simplify:

   $$\frac{x - 3}{\sqrt{x^{2} - 6 x + 13}}$$

The answer is:

$$\frac{x - 3}{\sqrt{x^{2} - 6 x + 13}}$$