d: \(x+y+\frac{2y^2}{x-y}\)

\(=\frac{\left(x+y\right)\left(x-y\right)+2y^2}{x-y}\)

\(=\frac{x^2-y^2+2y^2}{x-y}=\frac{x^2+y^2}{x-y}\)

\(\frac{x+y}{2xy}-\frac{1}{x+y}\)

\(=\frac{\left(x+y\right)^2-2xy}{2xy\left(x+y\right)}\)

\(=\frac{x^2+y^2+2xy-2xy}{2xy\left(x+y\right)}=\frac{x^2+y^2}{2xy\left(x+y\right)}\)

Ta có: \(\frac{x+y+\frac{2y^2}{x-y}}{\frac{x+y}{2xy}-\frac{1}{x+y}}\)

\(=\frac{x^2+y^2}{x-y}:\frac{x^2+y^2}{2xy\left(x+y\right)}=\frac{2xy\left(x+y\right)}{x-y}\)

c: \(\frac{x+1}{x-1}-\frac{x-1}{x+1}\)

\(=\frac{\left(x+1\right)^2-\left(x-1\right)^2}{\left(x-1\right)\left(x+1\right)}\)

\(=\frac{x^2+2x+1-x^2+2x-1}{\left(x-1\right)\left(x+1\right)}=\frac{4x}{\left(x-1\right)\left(x+1\right)}\)

\(1-\frac{x-1}{x+1}\)

\(=\frac{x+1-\left(x-1\right)}{x+1}=\frac{2}{x+1}\)

Ta có: \(\left(\frac{x+1}{x-1}-\frac{x-1}{x+1}\right):\left(1-\frac{x-1}{x+1}\right)\)

\(=\frac{4x}{\left(x-1\right)\left(x+1\right)}:\frac{2}{x+1}=\frac{4x}{\left(x-1\right)\left(x+1\right)}\cdot\frac{x+1}{2}=\frac{2x}{x-1}\)

b: \(\frac{x^2+y^2}{x}-y\)

\(=\frac{x^2+y^2-xy}{x}\)

\(\frac{1}{x}-\frac{1}{y}\)

\(=\frac{y-x}{xy}\)

Ta có: \(\frac{\frac{x^2+y^2}{x}-y}{\frac{1}{x}-\frac{1}{y}}\)

\(=\frac{x^2+y^2-xy}{x}:\frac{y-x}{xy}\)

\(=\frac{x^2-xy+y^2}{x}\cdot\frac{xy}{y-x}=\frac{y\left(x^2-xy+y^2\right)}{y-x}\)

a: \(\frac{x}{y}+\frac{y}{x}-2\)

\(=\frac{x^2+y^2-2xy}{xy}\)
\(=\frac{\left(x-y\right)^2}{xy}\)

\(\frac{x}{y}-\frac{y}{x}\)

\(=\frac{x^2-y^2}{xy}=\frac{\left(x-y\right)\left(x+y\right)}{xy}\)

Ta có: \(\left(\frac{x}{y}+\frac{y}{x}-2\right):\left(\frac{x}{y}-\frac{y}{x}\right)\)

\(=\frac{\left(x-y\right)^2}{xy}:\frac{\left(x-y\right)\left(x+y\right)}{xy}\)

\(=\frac{\left(x-y\right)^2}{xy}\cdot\frac{xy}{\left(x-y\right)\left(x+y\right)}=\frac{x-y}{x+y}\)