Derivative of (sqrt(x)-1)/(x)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = \sqrt{x} - 1$ and $g{\left(x \right)} = x$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

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

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

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

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

   To find $\frac{d}{d x} g{\left(x \right)}$:

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

   Now plug in to the quotient rule:

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

2. Now simplify:

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

The answer is:

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