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