Simplify the following expression as much as possible:

$$
\frac{(x+1)^{\frac{1}{2}}+1}{(x+1)^{\frac{1}{2}}-1} \cdot \frac{\left[(x+1)^{\frac{1}{2}}+1\right](x+1)^{-\frac{1}{2}}-\left[(x+1)^{\frac{1}{2}}-1\right](x+1)^{-\frac{1}{2}}}{2\left[(x+1)^{\frac{1}{2}}+1\right]^{2}}
$$
Let's simplify the fraction by \(\left[(x+1)^{\frac{1}{2}}+1\right]\) and factor out \((x+1)^{-\frac{1}{2}}\) from the numerator, yielding

$$
\begin{aligned}
\frac{(x+1)^{-\frac{1}{2}}\left[(x+1)^{\frac{1}{2}}+1-(x+1)^{\frac{1}{2}}+1\right]}{2\left[(x+1)^{\frac{1}{2}}\right]\left[-1(x+1)^{\frac{1}{2}}+1\right]}=\frac{2(x+1)^{-\frac{1}{2}}}{2[(x+1)-1]}=\frac{1}{x(x+1)^{\frac{1}{2}}} \\
\text { György Hoffmann (Bp. V., Eötvös g. II. o. t.) }
\end{aligned}
$$