\[\frac{x(x + 1)^{-\frac{1}{2}} - (x + 1)^{\frac{1}{2}}}{(x + 1)^{\frac{1}{2}}}\]

The numerator can be rewritten as: \[x(x + 1)^{-\frac{1}{2}} - (x + 1)^{\frac{1}{2}} = \frac{x}{(x + 1)^{\frac{1}{2}}} - (x + 1)^{\frac{1}{2}}\]

Combining the terms in the numerator over a common denominator: \[= \frac{x - (x + 1)}{(x + 1)^{\frac{1}{2}}}\]

\[= \frac{x - x - 1}{(x + 1)^{\frac{1}{2}}} = \frac{-1}{(x + 1)^{\frac{1}{2}}}\]

Substituting this back into the original expression gives:

\[\frac{\frac{-1}{(x + 1)^{\frac{1}{2}}}}{(x + 1)^{\frac{1}{2}}}\]

This simplifies to: \[\frac{-1}{(x + 1)^{\frac{1}{2}} \cdot (x + 1)^{\frac{1}{2}}} = \frac{-1}{(x + 1)}\]

 = \[{\frac{-1}{x + 1}}\]