\(=\dfrac{1}{x\left(x-y\right)\left(x-z\right)}-\dfrac{1}{y\left(y-z\right)\left(x-y\right)}+\dfrac{1}{z\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{yz\left(y-z\right)-xz\left(x-z\right)+xy\left(x-y\right)}{xyz\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{y^2z-yz^2-x^2z+xz^2+xy\left(x-y\right)}{xyz\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{z\left(y^2-x^2\right)+z^2\left(x-y\right)+xy\left(x-y\right)}{xyz\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)+xy\left(x-y\right)}{xyz\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{\left(x-y\right)\left(-zx-zy+z^2+xy\right)}{xyz\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{x\left(-z+y\right)+z\left(-y+z\right)}{xyz\left(x-z\right)\left(y-z\right)}\)
\(=\dfrac{\left(y-z\right)\left(x-z\right)}{xyz\left(x-z\right)\left(y-z\right)}=\dfrac{1}{xyz}\)
