Derivative of x-sqrt(x^2-4x+5)

1. Differentiate $x - \sqrt{\left(x^{2} - 4 x\right) + 5}$ 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 $u = \left(x^{2} - 4 x\right) + 5$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} - 4 x\right) + 5\right)$:

         1. Differentiate $\left(x^{2} - 4 x\right) + 5$ term by term:

            1. Differentiate $x^{2} - 4 x$ 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: $-4$

               The result is: $2 x - 4$

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

            The result is: $2 x - 4$

         The result of the chain rule is:

         $$\frac{2 x - 4}{2 \sqrt{\left(x^{2} - 4 x\right) + 5}}$$

      So, the result is: $- \frac{2 x - 4}{2 \sqrt{\left(x^{2} - 4 x\right) + 5}}$

   The result is: $- \frac{2 x - 4}{2 \sqrt{\left(x^{2} - 4 x\right) + 5}} + 1$

2. Now simplify:

   $$\frac{- x + \sqrt{x^{2} - 4 x + 5} + 2}{\sqrt{x^{2} - 4 x + 5}}$$

The answer is:

$$\frac{- x + \sqrt{x^{2} - 4 x + 5} + 2}{\sqrt{x^{2} - 4 x + 5}}$$