Partial Derivative/Examples/Notation for 3-Value Function

Examples of Notation for Partial Derivatives
Let $u = \map f {x, y, z}$ be a real function of $3$ variables.

Then the partial derivatives may be expressed variously as:


 * $\dfrac {\partial u} {\partial x} = \map {f_1} {x, y, z} = \dfrac {\partial f} {\partial x} = \map {\dfrac \partial {\partial x} f} {x, y, z}$


 * $\dfrac {\partial u} {\partial y} = \map {f_2} {x, y, z} = \dfrac {\partial f} {\partial y} = \map {\dfrac \partial {\partial y} f} {x, y, z}$


 * $\dfrac {\partial u} {\partial z} = \map {f_3} {x, y, z} = \dfrac {\partial f} {\partial z} = \map {\dfrac \partial {\partial z} f} {x, y, z}$