\(\frac{x}{\left(x-y\right)\left(x-z\right)}\) \(+\frac{y}{\left(x-y\right)\left(y-z\right)}\)\(+\frac{z}{\left(y-z\right)\left(z-x\right)}\)
\(=\)\(\frac{x\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\) \(+\frac{y\left(x-z\right)}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}-\)\(\frac{z\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\frac{x\left(y-z\right)+y\left(x-z\right)-z\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\)\(\frac{xy-xz+xy-yz-xz+yz}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\)\(\frac{2xy-2xz}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\frac{2x\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\)\(\frac{2x}{\left(x-y\right)\left(x-z\right)}\)